268:. The general case was reduced to two conjectures about the properties of the c-function and the so-called spherical Fourier transform. Explicit formulas for the c-function were later obtained for a large class of classical semisimple Lie groups by Bhanu-Murthy. In turn these formulas prompted Gindikin and Karpelevich to derive a product formula for the c-function, reducing the computation to Harish-Chandra's formula for the rank 1 case. Their work finally enabled Harish-Chandra to complete his proof of the Plancherel theorem for spherical functions in 1966.
23972:
13541:
27328:
18284:
13085:
23655:
13250:
17938:
15643:
19943:
7464:
12794:
23967:{\displaystyle c_{s_{\alpha }}(\lambda )=c_{0}{\frac {2^{-i(\lambda ,\alpha _{0})}\Gamma (i(\lambda ,\alpha _{0}))}{\Gamma \!\left({\frac {1}{2}}\left({\frac {1}{2}}m_{\alpha }+1+i(\lambda ,\alpha _{0})\right)\right)\Gamma \!\left({\frac {1}{2}}\left({\frac {1}{2}}m_{\alpha }+m_{2\alpha }+i(\lambda ,\alpha _{0})\right)\right)}},}
15387:
13536:{\displaystyle f(i)={1 \over 2\pi ^{2}}\int _{0}^{\infty }{\tilde {f}}(\lambda )\lambda \,d\lambda \int _{-\infty }^{\infty }{\sin \lambda t/2 \over \sinh t}\cosh {t \over 2}\,dt={1 \over 2\pi ^{2}}\int _{-\infty }^{\infty }{\tilde {f}}(\lambda ){\lambda \pi \over 2}\tanh \left({\pi \lambda \over 2}\right)\,d\lambda .}
14051:
17933:
19612:
11990:
12647:
7217:
9164:
7279:
10132:
14072:
odd can be treated directly by reduction to the usual
Fourier transform. The remaining real Lorentz groups can be deduced by Flensted-Jensen's method of descent, as can other semisimple Lie groups of real rank one. Flensted-Jensen's method of descent also applies to the treatment of real semisimple
25237:
found a short and simple proof, allowing the result to be deduced directly from versions of the Paley-Wiener and spherical inversion formula. He proved that the spherical transform of a Harish-Chandra
Schwartz function is a classical Schwartz function. His key observation was then to show that the
25131:
12347:
22190:
18279:{\displaystyle f(g)=M^{*}F(g)=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {F}}(2\lambda )M^{*}\Phi _{2\lambda }(g)2^{{\rm {dim}}\,A}|d(2\lambda )|^{2}\,d\lambda =\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )\varphi _{\lambda }(g)\,\,b(\lambda )2^{{\rm {dim}}\,A}|d(2\lambda )|^{2}\,d\lambda .}
3307:
10508:
14589:
6794:
20257:
6484:
3458:
7739:
9911:
18442:
3797:
22447:. Although it was originally introduced by Harish-Chandra in the asymptotic expansions of spherical functions, discussed above, it was also soon understood to be intimately related to intertwining operators between induced representations, first studied in this context by
19600:
14807:
21122:
13080:{\displaystyle \int _{-\infty }^{\infty }F'\!\left(x+{\frac {t^{2}}{2}}\right)\,dt=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f'\!\left(x+{\frac {t^{2}+u^{2}}{2}}\right)\,dt\,du={2\pi }\int _{0}^{\infty }f'\!\left(x+{\frac {r^{2}}{2}}\right)r\,dr=2\pi f(x).}
15856:
5281:
The second and third technique will be described below, with two different methods of descent: the classical one due
Hadamard, familiar from treatments of the heat equation and the wave equation on hyperbolic space; and Flensted-Jensen's method on the hyperboloid.
24884:
13898:
24110:
17744:
24291:
16330:
11850:
5950:
6098:
13241:
12785:
12461:
16194:
7078:
7564:
10637:
18936:
8932:
24604:
15064:
20667:
9940:
2418:
21276:
24933:
12177:
9758:
18740:
8774:
13870:
6349:
3159:
15638:{\displaystyle F(g)={1 \over |W|}\int _{{\mathfrak {a}}^{*}}{\tilde {F}}(\lambda )\Phi _{\lambda }(g)|d(\lambda )|^{2}\,d\,\lambda =\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {F}}(\lambda )\Phi _{\lambda }(g)|d(\lambda )|^{2}\,d\,\lambda ,}
8445:
271:
In many special cases, for example for complex semisimple group or the
Lorentz groups, there are simple methods to develop the theory directly. Certain subgroups of these groups can be treated by techniques generalising the well-known
3001:
10329:
17622:
14456:
12452:
6648:
888:
6204:
22029:
20938:
20099:
20088:
19938:{\displaystyle c(\lambda )=c_{0}\cdot \prod _{\alpha \in \Sigma _{0}^{+}}{\frac {2^{-i(\lambda ,\alpha _{0})}\Gamma (i(\lambda ,\alpha _{0}))}{\Gamma \!\left({\frac {1}{2}}\left\right)\Gamma \!\left({\frac {1}{2}}\left\right)}}}
8377:
6356:
3320:
23429:
15251:
7584:
22420:
21690:
9479:
22824:
5524:
3898:
19453:
5248:
1060:
22944:
8563:
8080:
5681:
4461:
4075:
22018:
21012:
9767:
18315:
5812:
3646:
22724:
21553:
15378:
11688:
11078:
11469:
2831:
2635:
17388:
8877:
8276:
7459:{\displaystyle \int _{{\mathfrak {H}}^{2}}f_{1}{\overline {f_{2}}}\,dA=\int _{-\infty }^{\infty }{\tilde {f}}_{1}{\overline {\tilde {f_{2}}}}{\lambda \pi \over 2}\tanh \left({\pi \lambda \over 2}\right)\,d\lambda .}
4271:
13772:
12170:
3626:
7046:
19464:
14670:
14349:
14240:
21035:
12080:
6956:
6619:
11823:
11606:
5033:
17706:
1220:
15718:
2902:
2543:
24764:
20574:
8182:
5155:
4663:
4177:
1986:
15915:
23291:
2235:
23977:
1729:
23522:
17178:
16794:
23034:
24168:
19035:
11273:
9643:
1923:
20001:
16211:
14632:
5821:
25187:
5972:
20805:
11737:
7833:
6851:
4891:
4596:
10304:
6261:
2737:
13131:
10819:
10732:
1558:
22292:
16083:
15154:
18661:
18616:
11537:
11359:
7481:
5261:
107:
25645:
24759:
21363:
10515:
18981:
18828:
17262:
16639:
7902:
5426:
1812:
26040:
Anker, Jean-Philippe (1991), "The spherical
Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi, and Varadarajan",
23205:
17094:
16853:
15103:
12668:
10944:
3145:
17449:
24491:
14920:
23586:
20584:
16930:
13634:
2311:
21181:
20313:
5267:
variants of
Hadamard's method of descent, realising 2-dimensional hyperbolic space as the quotient of 3-dimensional hyperbolic space by the free action of a 1-parameter subgroup of SL(2,
3516:
24423:
20752:
9652:
8612:
4510:
4328:
18668:
9241:
559:
23138:
2125:
24459:
23099:
21799:
19292:
17033:
16056:
15681:
8677:
4748:
2067:
1638:
18551:
17502:
16569:
15974:
6533:
3054:
13781:
6266:
2299:
25229:
21888:
1353:
24380:
22573:
22329:
21854:
21725:
21598:
21403:
19338:
19256:
19158:
18821:
14663:
14445:
13667:
10225:
8659:
7952:
7250:
6884:
5563:
5331:
5066:
4789:
3947:
2451:
14046:{\displaystyle f(w)={1 \over \pi ^{2}}\int _{0}^{\infty }{\tilde {f}}(\lambda )\varphi _{\lambda }(w){\lambda \pi \over 2}\tanh \left({\pi \lambda \over 2}\right)\,d\lambda .}
11187:
2911:
8925:
4948:
14279:
25396:
24918:
22218:
21458:
21154:
20829:
20698:
20478:
20442:
20398:
19182:
19127:
19103:
19067:
18790:
18766:
18575:
18516:
18472:
17518:
16519:
15713:
14894:
14404:
14190:
14166:
14118:
12360:
939:
791:
767:
680:
595:
509:
477:
442:
17928:{\displaystyle f(g)=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )\varphi _{\lambda }(g)\,\,2^{{\rm {dim}}\,A}\cdot |b(\lambda )|\cdot |d(2\lambda )|^{2}\,d\lambda ,}
6111:
20846:
20342:
20021:
14866:
13120:
11985:{\displaystyle f(x)=\int _{-\infty }^{\infty }\varphi _{\lambda }(x){\tilde {f}}(\lambda )\,{\lambda \pi \over 2}\tanh \left({\frac {\pi \lambda }{2}}\right)\,d\lambda ,}
3545:
24670:
23316:
15161:
16488:
295:
combines the
Fourier inversion formula on a vector group with summation over a cocompact lattice. In Selberg's analogue of this formula, the vector group is replaced by
21603:
9541:
9325:
24347:
about the origin if the Paley-Wiener estimate is satisfied. This follows because the integrand defining the inverse transform extends to a meromorphic function on the
22729:
12642:{\displaystyle {\tilde {f}}(\lambda )=\int _{0}^{\infty }F\!\left({\frac {a^{2}+a^{-2}}{2}}\right)a^{-i\lambda }\,da=\int _{0}^{\infty }F(\cosh t)e^{-it\lambda }\,dt.}
5435:
3826:
19363:
7212:{\displaystyle f(x)=\int _{-\infty }^{\infty }\varphi _{\lambda }(x){\tilde {f}}(\lambda ){\lambda \pi \over 2}\tanh \left({\pi \lambda \over 2}\right)\,d\lambda ,}
5162:
22831:
8452:
7959:
5598:
4350:
3954:
24336:
noticed that the Paley-Wiener theorem and the spherical inversion theorem could be proved simultaneously, by a trick which considerably simplified previous proofs.
20943:
22338:
5697:
22610:
21481:
15291:
11611:
10953:
9159:{\displaystyle g_{t}\cdot A={\begin{pmatrix}a\cosh 2t+y\sinh 2t+b&x+i(y\cosh 2t+a\sinh 2t)\\x-i(y\cosh 2t+a\sinh 2t)&a\cosh 2t+y\sinh 2t-b\end{pmatrix}}.}
8382:
11364:
2762:
2550:
17277:
8793:
8191:
4186:
21926:
13697:
3563:
10127:{\displaystyle \int _{H^{3}}F\,dV=4\pi \int _{-\infty }^{\infty }F(t)\sinh ^{2}t\,dt,\,\,\,\int _{H^{2}}f\,dV=2\pi \int _{-\infty }^{\infty }f(t)\sinh t\,dt.}
6965:
14286:
25126:{\displaystyle {\mathcal {S}}(K\backslash G/K)=\left\{f\in C^{\infty }(G/K)^{K}:\sup _{x}\left|(1+d(x,o))^{m}(\Delta +I)^{n}f(x)\right|<\infty \right\}.}
12342:{\displaystyle {\tilde {f}}(\lambda )=\int _{-\infty }^{\infty }\int _{0}^{\infty }f\!\left({\frac {a^{2}+a^{-2}+b^{2}}{2}}\right)a^{-i\lambda /2}\,da\,db,}
6891:
6552:
11744:
11546:
8287:
12085:
4960:
17629:
2840:
27217:
12006:
5257:
3302:{\displaystyle \int _{G}f_{1}{\overline {f_{2}}}\,dg=\int {\tilde {f}}_{1}(\lambda ){\overline {{\tilde {f}}_{2}(\lambda )}}\,\lambda ^{2}\,d\lambda .}
2479:
226:
20483:
8089:
5094:
4609:
4084:
1932:
974:
23216:
14124:
in terms of his c-function, discussed below. Although less general, it gives a simpler approach to the
Plancherel theorem for this class of groups.
2139:
1136:
26767:
23638:
1643:
23434:
10503:{\displaystyle (-\Delta _{2}+R_{2})f=\partial _{t}^{2}f+\coth t\partial _{t}f+r\partial _{r}f=\partial _{t}^{2}f+(\coth t+\tanh t)\partial _{t}f.}
22949:
14584:{\displaystyle \chi _{\lambda }(e^{X})=\operatorname {Tr} \pi _{\lambda }(e^{X}),(X\in {\mathfrak {t}}),\qquad d(\lambda )=\dim \pi _{\lambda }.}
6789:{\displaystyle \varphi _{\lambda }(e^{t}i)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left(\cosh t-\sinh t\cos \theta \right)^{-1-i\lambda }\,d\theta .}
27053:
11194:
9555:
1855:
22185:{\displaystyle \varphi _{\lambda }(e^{X})=e^{i\lambda -\rho }\int _{\sigma (N)}{{\overline {\lambda '(n)}} \over \lambda '(e^{X}ne^{-X})}\,dn,}
14096:
applies to a wide class of real semisimple Lie groups of arbitrary real rank and yields the explicit product form of the
Plancherel measure on
20252:{\displaystyle W:f\mapsto {\tilde {f}},\,\,\,\ L^{2}(K\backslash G/K)\rightarrow L^{2}({\mathfrak {a}}_{+}^{*},|c(\lambda )|^{-2}\,d\lambda )}
6479:{\displaystyle b(\lambda )=f_{\lambda }(i)=\int {\sin \lambda t \over \lambda \sinh t}\,dt={\pi \over \lambda }\tanh {\pi \lambda \over 2},}
5087:
determined up to a sign. The
Laplacian has the following form on functions invariant under SO(2), regarded as functions of the real parameter
3453:{\displaystyle {\begin{cases}U:L^{2}(K\backslash G/K)\to L^{2}(\mathbb {R} ,\lambda ^{2}\,d\lambda )\\U:f\longmapsto {\tilde {f}}\end{cases}}}
2472:
determined up to a sign. The Laplacian has the following form on functions invariant under SU(2), regarded as functions of the real parameter
26880:
21910:
whose complement is a union of cells of strictly smaller dimension and therefore has measure zero. It follows that the integral formula for φ
20757:
14195:
11693:
7764:
7734:{\displaystyle {\mathcal {S}}=\left\{f\left|\sup _{t}\left|(1+t^{2})^{N}(I+\Delta )^{M}f(t)\varphi _{0}(t)\right|<\infty \right\}\right..}
6803:
4811:
4537:
10234:
6211:
2668:
27352:
25733:. The integral then becomes an alternating sum of multidimensional Godement-type integrals, whose combinatorics is governed by that of the
10752:
10660:
1477:
22226:
11476:
11282:
303:, the Fourier transform by the spherical transform and the lattice by a cocompact (or cofinite) discrete subgroup. The original paper of
24677:
21292:
284:
gave a general method for deducing properties of the spherical transform for a real semisimple group from that of its complexification.
27043:
17183:
16576:
9906:{\displaystyle L_{3}=\partial _{b}^{2}+\partial _{x}^{2}+\partial _{y}^{2},\,\,\,R_{3}=b\partial _{b}+x\partial _{x}+y\partial _{y}.\,}
5338:
1748:
23149:
18437:{\displaystyle b(\lambda )=C\cdot d(2\lambda )^{-1}\cdot \prod _{\alpha >0}\tanh {\pi (\alpha ,\lambda ) \over (\alpha ,\alpha )},}
10883:
3792:{\displaystyle {\mathcal {S}}=\left\{f\left|\sup _{t}\left|(1+t^{2})^{N}(I+\Delta )^{M}f(t)\sinh(t)\right|<\infty \right\}\right..}
17395:
15861:
5252:
There are several methods for deriving the corresponding eigenfunction expansion for this ordinary differential equation including:
27362:
27170:
27025:
16858:
13562:
353:
26800:(1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series",
17107:
16723:
27001:
26156:
Flensted-Jensen, Mogens (1978), "Spherical functions of a real semisimple Lie group. A method of reduction to the complex case",
147:
were due to Segal and Mautner. At around the same time, Harish-Chandra and Gelfand & Naimark derived an explicit formula for
20703:
19595:{\displaystyle f(g)=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )\varphi _{\lambda }(g)\,|c(\lambda )|^{-2}\,d\lambda ,}
18990:
14802:{\displaystyle \chi _{\lambda }(e^{X})={\sum _{\sigma \in W}{\rm {sign}}(\sigma )e^{i\lambda (\sigma X)} \over \delta (e^{X})},}
961:
27357:
21117:{\displaystyle \Delta \varphi _{\lambda }=\left(\left\|\lambda \right\|^{2}+\left\|\rho \right\|^{2}\right)\varphi _{\lambda }}
19948:
14596:
26755:
L'analyse harmonique sur les espaces symétriques de rang 1. Une réduction aux espaces hyperboliques réels de dimension impaire
25138:
26815:
21753:
16993:
4672:
1991:
1571:
774:
179:
16524:
25734:
15851:{\displaystyle d(\lambda )\delta (e^{X})\Phi _{\lambda }(e^{X})=\sum _{\sigma \in W}{\rm {sign}}(\sigma )e^{i\lambda (X)},}
8626:
Hadamard's method of descent relied on functions invariant under the action of 1-parameter subgroup of translations in the
2244:
26683:
26532:
15115:
1289:
24879:{\displaystyle Cf=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )\varphi _{\lambda }|c(\lambda )|^{-2}\,d\lambda .}
24618:. A further estimate for the integral shows that it is in fact given by a measure and that therefore there is a constant
18625:
18580:
10161:
25593:
11124:
7841:), by verifying directly that the spherical transform satisfies the given growth conditions and then using the relation
26893:
26762:
24325:
independently of the spherical inversion theorem, using a modification of his method of reduction to the complex case.
18943:
8882:
7844:
4900:
24105:{\displaystyle c_{0}=2^{m_{\alpha }/2+m_{2\alpha }}\Gamma \!\left({\tfrac {1}{2}}(m_{\alpha }+m_{2\alpha }+1)\right).}
23644:
by directly computing the integral, which can be expressed as an integral in two variables and hence a product of two
17044:
16803:
15071:
14872:) is given by a product formula (Weyl's denominator formula) which extends holomorphically to the complexification of
3080:
26982:
26873:
26744:
26723:
26670:
26649:
26600:
26582:
26308:
26185:
26147:
26126:
22440:
16372:, complex linear on its Lie algebra, chosen to commute with σ. The fixed point subgroup of τσ is a compact real form
14362:
37:
26062:
Beerends, R. J. (1987), "The Fourier transform of Harish-Chandra's c-function and inversion of the Abel transform",
25964:
The second statement on supports follows from Flensted-Jensen's proof by using the explicit methods associated with
24339:
The first step of his proof consists in showing directly that the inverse transform, defined using Harish-Chandra's
27252:
24286:{\displaystyle |{\tilde {f}}(\lambda )|\leq C_{N}(1+|\lambda |)^{-N}e^{R\left|\operatorname {Im} \lambda \right|}.}
24115:
23557:
26765:(1977), "A quick proof of Harish-Chandra's Plancherel theorem for spherical functions on a semisimple Lie group",
16325:{\displaystyle f(u)=\sum _{\lambda }{\tilde {f}}(\lambda ){\chi _{\lambda }(u) \over d(\lambda )}d(\lambda )^{2}.}
5945:{\displaystyle \int _{{\mathfrak {H}}^{3}}(M_{1}f)\cdot F\,dV=\int _{{\mathfrak {H}}^{2}}f\cdot (M_{1}^{*}F)\,dA.}
26897:
26469:
26431:
24625:
22520:), which can be identified with the representation induced from the 1-dimensional representation defined by λ on
20262:
6093:{\displaystyle \int _{{\mathfrak {H}}^{3}}(M_{0}f)\cdot F\,dV=\int _{{\mathfrak {H}}^{2}}f\cdot (M_{0}^{*}F)\,dA}
3465:
1406:
24385:
16451:
16350:
14074:
8570:
4468:
4286:
22430:
9174:
533:
23104:
13236:{\displaystyle F(\cosh t)={2 \over \pi }\int _{0}^{\infty }{\tilde {f}}(i\lambda )\cos(\lambda t)\,d\lambda .}
2082:
27048:
24428:
23068:
19261:
16189:{\displaystyle {\tilde {f}}(\lambda )=\int _{U}f(u){{\overline {\chi _{\lambda }(u)}} \over d(\lambda )}\,du}
16025:
15650:
18525:
17467:
15924:
7559:{\displaystyle \Delta _{2}\varphi _{\lambda }=\left(\lambda ^{2}+{\tfrac {1}{4}}\right)\varphi _{\lambda }.}
6491:
3008:
27331:
27104:
27038:
26866:
25408:
21374:(λ). In particular they are uniquely determined and the series and its derivatives converges absolutely on
17454:
10632:{\displaystyle \partial _{t}^{2}+(\coth t+\tanh t)\partial _{t}=\partial _{t}^{2}+2\coth(2t)\partial _{t}.}
26573:
Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions
25200:
22828:
The key property of the intertwining operators and their integrals is the multiplicative cocycle property
21859:
18931:{\displaystyle {\mathfrak {g}}_{\alpha }=\{X\in {\mathfrak {g}}:=\alpha (H)X\,\,(H\in {\mathfrak {a}})\}.}
27068:
24354:
22547:
22303:
21828:
21699:
21572:
21377:
19312:
19195:
19132:
18795:
14637:
14419:
13641:
12780:{\displaystyle f(x)={\frac {-1}{2\pi }}\int _{-\infty }^{\infty }F'\!\left(x+{t^{2} \over 2}\right)\,dt.}
8633:
7926:
7224:
6858:
5537:
5305:
5040:
4763:
3921:
2425:
613:
23309:). The cocycle property of the intertwining operators implies a similar multiplicative property for the
27313:
27267:
27191:
27073:
26248:
Gindikin, S.G. (2008), "Horospherical transform on Riemannian symmetric manifolds of noncompact type",
26203:
24599:{\displaystyle T(f)=\int _{{\mathfrak {a}}_{+}^{*}}{\tilde {f}}(\lambda )|c(\lambda )|^{-2}\,d\lambda }
15059:{\displaystyle \int _{G}F(g)\,dg={1 \over |W|}\int _{\mathfrak {a}}F(e^{X})\,|\delta (e^{X})|^{2}\,dX.}
1820:
proved, the representations of the spherical principal series are irreducible and two representations π
288:
84:
23065:, namely the number of positive restricted roots, is the unique element that carries the Weyl chamber
20662:{\displaystyle L=\Delta _{\mathfrak {a}}-\sum _{\alpha >0}m_{\alpha }\,\coth \alpha \,A_{\alpha },}
14257:
27308:
27124:
25412:
25377:
24899:
22199:
21439:
21135:
20810:
20679:
20459:
20423:
20379:
19163:
19108:
19084:
19048:
18771:
18747:
18556:
18497:
18453:
16500:
16010:
is the diagonal subgroup. The spherical functions for the latter space, which can be identified with
15694:
15688:
14875:
14171:
14147:
14099:
13123:
2746:
2413:{\displaystyle \Delta =-r^{2}(\partial _{x}^{2}+\partial _{y}^{2}+\partial _{r}^{2})+r\partial _{r}.}
920:
748:
661:
576:
490:
458:
423:
292:
45:
21271:{\displaystyle f_{\lambda }=e^{i\lambda -\rho }\sum _{\mu \in \Lambda }a_{\mu }(\lambda )e^{-\mu },}
3329:
27160:
27058:
26961:
9753:{\displaystyle L_{2}=\partial _{b}^{2}+\partial _{x}^{2},\,\,\,R_{2}=b\partial _{b}+x\partial _{x}}
777:
345:
324:
182:
168:
57:
21278:
with Λ the cone of all non-negative integer combinations of positive roots, and the transforms of
20318:
18735:{\displaystyle {\mathfrak {g}}={\mathfrak {k}}+{\mathfrak {p}},\,\,G=\exp {\mathfrak {p}}\cdot K.}
18522:
be a maximal compact subgroup given as the subgroup of fixed points of a Cartan involution σ. Let
14821:
13096:
4601:
3521:
1926:
417:. The closure of its (the Hecke algebra's) image in the operator norm is a non-unital commutative
27257:
27033:
26537:
25238:
inverse transform was continuous on the Paley-Wiener space endowed with classical Schwartz space
23630:
1372:
1119:
683:
214:
111:
8769:{\displaystyle g_{t}={\begin{pmatrix}\cosh t&i\sinh t\\-i\sinh t&\cosh t\end{pmatrix}}.}
1245:
27288:
27232:
27196:
26180:, CBMS regional conference series in mathematics, vol. 61, American Mathematical Society,
12660:
9514:
1447:
562:
334:
88:
26837:
Takahashi, R. (1963), "Sur les représentations unitaires des groupes de Lorentz généralisés",
13865:{\displaystyle {\tilde {f}}_{1}(\lambda )={\tilde {f}}(\lambda )\cdot \varphi _{\lambda }(w),}
6344:{\displaystyle \Delta _{2}f_{\lambda }=\left(\lambda ^{2}+{\tfrac {1}{4}}\right)f_{\lambda }.}
26995:
8440:{\textstyle {\frac {\lambda \pi }{2}}\tanh \left({\frac {\pi \lambda }{2}}\right)\,d\lambda }
1097:
628:
311:
who brought the transform to the fore, giving in particular an elementary treatment for SL(2,
253:
141:
53:
26991:
26284:
27271:
26460:
26334:
26240:
26220:
26107:
24140:
23634:
21743:
14354:
7752:
3814:
3802:
2996:{\displaystyle f(x)=\int {\tilde {f}}(\lambda )\Phi _{\lambda }(x)\lambda ^{2}\,d\lambda .}
202:
152:
64:
for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the
29:
25:
26858:
23038:
for the length function on the Weyl group associated with the choice of Weyl chamber. For
8:
27237:
27175:
26889:
484:
445:
26338:
26227:[Plancherel measure for symmetric Riemannian spaces of non-positive curvature],
17617:{\displaystyle b(\lambda )=M^{*}\Phi _{2\lambda }(1)=\int _{K}\Phi _{2\lambda }(k)\,dk.}
12447:{\displaystyle F(u)=\int _{-\infty }^{\infty }f\!\left(u+{\frac {t^{2}}{2}}\right)\,dt,}
883:{\displaystyle \chi _{\lambda }(\pi (f))=\int _{G}f(g)\cdot \varphi _{\lambda }(g)\,dg.}
159:. A simpler abstract formula was derived by Mautner for a "topological" symmetric space
27262:
27129:
26786:
26571:
26520:
26486:
26448:
26413:
26400:
26365:
26352:
26265:
26079:
25965:
21406:
16334:
There is an evident duality between these formulas and those for the non-compact dual.
15267:
corresponding to λ; these representations are irreducible and can all be realized on L(
9547:
6199:{\displaystyle M_{i}^{*}\Delta _{3}=\left(\Delta _{2}+{\tfrac {3}{4}}\right)M_{i}^{*}.}
4953:
2742:
2304:
703:
218:
69:
61:
41:
26781:
25260:, pp. 492–493, historical notes on the Plancherel theorem for spherical functions
20933:{\displaystyle L=L_{0}-\sum _{\alpha >0}m_{\alpha }\,(\coth \alpha -1)A_{\alpha },}
27242:
26740:
26719:
26666:
26645:
26596:
26578:
26551:
26535:(1970), "A duality for symmetric spaces with applications to group representations",
26524:
26418:
26370:
26304:
26269:
26181:
26169:
26143:
26122:
26091:
26083:
26054:
25648:
25194:
24122:(λ) is an immediate consequence of this formula and the multiplicative properties of
20083:{\displaystyle \rho ={\frac {1}{2}}\sum _{\alpha \in \Sigma ^{+}}m_{\alpha }\alpha .}
15918:
8372:{\displaystyle W:L^{2}({\mathfrak {H}}^{2})\to L^{2}([0,\infty )\times \mathbb {R} )}
8284:
are irreducible and unitarily equivalent only when the sign of λ is changed. The map
7837:
Both these results can be deduced by descent from the corresponding results for SL(2,
7568:
4804:
4279:
are irreducible and unitarily equivalent only when the sign of λ is changed. The map
3630:
2132:
273:
49:
26703:
26135:
23424:{\displaystyle c_{s_{1}s_{2}}(\lambda )=c_{s_{1}}(s_{2}\lambda )c_{s_{2}}(\lambda )}
15246:{\displaystyle \Phi _{\lambda }(e^{X})={\chi _{\lambda }(e^{X}) \over d(\lambda )}.}
137:
The first versions of an abstract Plancherel formula for the Fourier transform on a
27247:
27165:
27134:
27114:
27099:
27094:
27089:
26846:
26824:
26776:
26698:
26658:
26625:
26546:
26510:
26478:
26440:
26408:
26390:
26360:
26342:
26296:
26292:
26257:
26165:
26071:
26049:
25542:
These are indexed by highest weights shifted by half the sum of the positive roots.
25366:
The spectrum coincides with that of the commutative Banach *-algebra of integrable
24348:
14137:
277:
265:
186:
138:
103:
26926:
26498:
21685:{\displaystyle \varphi _{\lambda }(e^{t}X)\sim c(\lambda )e^{(i\lambda -\rho )Xt}}
15255:
They are the matrix coefficients of the irreducible spherical principal series of
9474:{\displaystyle dV=(1+r^{2})^{-1/2}\,db\,dx\,dy,\,\,\,dA=(1+r^{2})^{-1/2}\,db\,dx,}
287:
One of the principal applications and motivations for the spherical transform was
27109:
27063:
27011:
27006:
26977:
26679:
26456:
26381:
Harish-Chandra (1952), "Plancherel formula for the 2 x 2 real unimodular group",
26236:
26225:Мера Планшереля для римановых симметрических пространств неположительной кривизны
26114:
26103:
25749:
22819:{\displaystyle A(s,\lambda )\pi _{\lambda }(g)=\pi _{s\lambda }(g)A(s,\lambda ).}
21812:
17464:. It follows from the eigenfunction characterisation of spherical functions that
15980:
8615:
7744:
5519:{\displaystyle \Delta _{3}M_{1}f=M_{1}\left(\Delta _{2}+{\tfrac {3}{4}}\right)f,}
4513:
3893:{\displaystyle |F(\lambda )|\leq Ce^{R\left|\operatorname {Im} \lambda \right|}.}
3806:
1828:
are unitarily equivalent if and only if μ = σ(λ) for some σ in the Weyl group of
632:
624:
410:
248:λ as the Plancherel measure. He verified this formula for the special cases when
210:
80:
73:
65:
26936:
19448:{\displaystyle {\tilde {f}}(\lambda )=\int _{G}f(g)\varphi _{-\lambda }(g)\,dg.}
5243:{\displaystyle \int f\,dA=\int _{-\infty }^{\infty }f(t)\left|\sinh t\right|dt.}
1055:{\displaystyle \varphi _{\lambda }(g)=\int _{K}\lambda ^{\prime }(gk)^{-1}\,dk.}
27298:
27150:
26951:
26629:
26595:, Pure and Applied Mathematics 80, New York: Academic Press, pp. xvi+628,
26322:
26276:
26216:
26194:
24474:
22939:{\displaystyle A(s_{1}s_{2},\lambda )=A(s_{1},s_{2}\lambda )A(s_{2},\lambda ),}
18479:
16199:
16071:
15260:
8558:{\displaystyle Wf(\lambda ,x)=\int _{G/K}f(g)\pi _{\lambda }(g)\xi _{0}(x)\,dg}
8075:{\displaystyle \pi _{\lambda }(g^{-1})\xi (x)=|cx+d|^{-1-i\lambda }\xi (g(x)).}
5676:{\displaystyle \Delta _{3}M_{0}=M_{0}\left(\Delta _{2}+{\tfrac {3}{4}}\right).}
4456:{\displaystyle Wf(\lambda ,z)=\int _{G/K}f(g)\pi _{\lambda }(g)\xi _{0}(z)\,dg}
4070:{\displaystyle \pi _{\lambda }(g^{-1})\xi (z)=|cz+d|^{-2-i\lambda }\xi (g(z)).}
1123:
175:
33:
26829:
26261:
26201:(1948), "An analog of Plancherel's formula for the complex unimodular group",
23046:, this is the number of chambers crossed by the straight line segment between
21007:{\displaystyle L_{0}=\Delta _{\mathfrak {a}}-\sum _{\alpha >0}A_{\alpha },}
16357:, the fixed points of an involution σ, conjugate linear on the Lie algebra of
27346:
27303:
27227:
26956:
26941:
26931:
26813:
Stade, E. (1999), "The hyperbolic tangent and generalized Mellin inversion",
24151:. It is a necessary and sufficient condition that the spherical transform be
23645:
23546:, the reflection in a (simple) root α, the so-called "rank-one reduction" of
22415:{\displaystyle c(\lambda )=\int _{\sigma (N)}{\overline {\lambda '(n)}}\,dn.}
14247:
14061:
7473:
5807:{\displaystyle M_{1}^{*}F(x,r)=r^{1/2}\int _{-\infty }^{\infty }F(x,y,r)\,dy}
3555:
448:
can be identified with the continuous functions vanishing at infinity on its
383:
156:
119:
26467:
Harish-Chandra (1958b), "Spherical Functions on a Semisimple Lie Group II",
26429:
Harish-Chandra (1958a), "Spherical functions on a semisimple Lie group. I",
22719:{\displaystyle A(s,\lambda )F(k)=\int _{\sigma (N)\cap s^{-1}Ns}F(ksn)\,dn,}
21548:{\displaystyle \varphi _{\lambda }=\sum _{s\in W}c(s\lambda )f_{s\lambda }.}
18494:
be a noncompact connected real semisimple Lie group with finite center. Let
15373:{\displaystyle {\tilde {F}}(\lambda )=\int _{G}F(g)\Phi _{-\lambda }(g)\,dg}
11683:{\displaystyle b(\lambda )=M^{*}\Phi _{2\lambda }(0)=\pi \tanh \pi \lambda }
11073:{\displaystyle \int _{H^{3}}F\,dV=\int _{H^{2}}(1+b^{2}+x^{2})^{1/2}QF\,dA.}
125:
The main reference for almost all this material is the encyclopedic text of
27293:
26946:
26916:
26797:
26739:, Annals of Mathematics Studies, vol. 87, Princeton University Press,
26422:
26374:
26347:
26198:
23633:
for which the asymptotic expansion is known from the classical formulas of
23140:. By Harish-Chandra's integral formula, it corresponds to Harish-Chandra's
19078:
15684:
14448:
11464:{\displaystyle \int _{H^{3}}(Mf)\cdot F\,dV=\int _{H^{2}}f\cdot (M*F)\,dV.}
6636:
2826:{\displaystyle \Phi _{\lambda }(t)={\sin \lambda t \over \lambda \sinh t},}
2630:{\displaystyle \int f\,dV=\int _{-\infty }^{\infty }f(t)\,\sinh ^{2}t\,dt.}
942:
222:
26684:"On the existence and irreducibility of certain series of representations"
26395:
17453:
There is a similar compatibility for other operators in the center of the
17383:{\displaystyle \int _{G/U}(Mf)\cdot F=\int _{G_{0}/K_{0}}f\cdot (M^{*}F).}
8872:{\displaystyle A={\begin{pmatrix}a+b&x+iy\\x-iy&a-b\end{pmatrix}}}
8271:{\displaystyle \Phi _{\lambda }(g)=(\pi _{\lambda }(g)\xi _{0},\xi _{0}).}
4266:{\displaystyle \Phi _{\lambda }(g)=(\pi _{\lambda }(g)\xi _{0},\xi _{0}).}
27222:
27212:
27119:
26921:
26663:
Representation theory of semisimple groups: an overview based on examples
24481:
large and hence can be shown to vanish outside the closed ball of radius
22726:
where σ is the Cartan involution. It satisfies the intertwining relation
22013:{\displaystyle \varphi _{\lambda }(g)=\int _{K/M}\lambda '(gk)^{-1}\,dk.}
19185:
18447:
13767:{\displaystyle \pi _{\lambda }(f)\xi _{0}={\tilde {f}}(\lambda )\xi _{0}}
9169:
8665:) which splits as a direct product of SO(2) and the 1-parameter subgroup
7915:
is the matrix coefficient of the spherical principal series defined on L(
3910:
is the matrix coefficient of the spherical principal series defined on L(
3621:{\displaystyle \Delta \Phi _{\lambda }=(\lambda ^{2}+1)\Phi _{\lambda }.}
1428:
1368:
414:
17:
18308:(λ) was known from the prior determination of the Plancherel measure by
7041:{\displaystyle {(M_{1}^{*}F)}^{\sim }(\lambda )={\tilde {F}}(\lambda ).}
1466:
and the character λ. The induced representation is defined on functions
27155:
26987:
26851:
26790:
26711:
26637:
26515:
26490:
26452:
26075:
25233:
The original proof of Harish-Chandra was a long argument by induction.
21018:
can be regarded as a perturbation of the constant-coefficient operator
19189:
14816:
14344:{\displaystyle A=\exp i{\mathfrak {t}},\qquad P=\exp i{\mathfrak {u}},}
14235:{\displaystyle {\mathfrak {g}}={\mathfrak {u}}\oplus i{\mathfrak {u}}.}
14120:* without using Harish-Chandra's expansion of the spherical functions φ
9288:= 0 can be identified with the orbit of the identity matrix under SL(2,
8661:. Flensted–Jensen's method uses the centraliser of SO(2) in SL(2,
6635:
of the ordinary differential equation must vanish or by expanding as a
1849:
1356:
1280:
418:
48:
in the representation theory of the group of real numbers in classical
26281:
Introduction aux travaux de A. Selberg (Exposé no. 144, February 1957)
24896:
to be deduced from the Plancherel theorem and Paley-Wiener theorem on
23058:
in the interior of the chamber. The unique element of greatest length
20480:, it defines a first order differential operator (or vector field) by
6951:{\displaystyle {\tilde {f}}(\lambda )=\int f\varphi _{-\lambda }\,dA,}
6614:{\displaystyle \phi _{\lambda }=b(\lambda )^{-1}M_{1}\Phi _{\lambda }}
26732:
26404:
26356:
24343:-function, defines a function supported in the closed ball of radius
11818:{\displaystyle {(M^{*}F)}^{\sim }(\lambda )={\tilde {F}}(2\lambda ).}
11601:{\displaystyle M^{*}\Phi _{2\lambda }=b(\lambda )\varphi _{\lambda }}
9260:. Similarly the action of SO(2) acts by rotation on the coordinates (
6632:
519:
337:
236:(λ) to describe the asymptotic behaviour of the spherical functions φ
26482:
26444:
22503:, it is an eigenvector of the intertwining operator with eigenvalue
19184:. With respect to this inner product, the restricted roots Σ give a
12165:{\displaystyle {\tilde {f}}(\lambda )=\int _{S}f(s)\alpha '(s)\,ds,}
5028:{\displaystyle \Delta =-r^{2}(\partial _{x}^{2}+\partial _{r}^{2}).}
455:. Points in the spectrum are given by continuous *-homomorphisms of
25239:
23550:. In fact the integral involves only the closed connected subgroup
17701:{\displaystyle (M^{*}F)^{\sim }(\lambda )={\tilde {F}}(2\lambda ).}
16198:
and the spherical inversion formula now follows from the theory of
2639:
Identifying the square integrable SU(2)-invariant functions with L(
449:
26757:, Lecture Notes in Math, vol. 739, Springer, pp. 623–646
3805:, the spherical transforms of smooth SU(2)-invariant functions of
2897:{\displaystyle {\tilde {f}}(\lambda )=\int f\Phi _{-\lambda }\,dV}
12075:{\displaystyle \varphi _{\lambda }(g)=\int _{K}\alpha '(kg)\,dk,}
7221:
the spherical inversion formula for SO(2)-invariant functions on
2538:{\displaystyle \Delta =-\partial _{t}^{2}-2\coth t\partial _{t}.}
148:
26325:(1951), "Plancherel formula for complex semisimple Lie groups",
21174:
with the same eigenvalue, it is natural look for a formula for φ
20569:{\displaystyle Xf(y)=\left.{\frac {d}{dt}}f(y+tX)\right|_{t=0}.}
20347:
8177:{\displaystyle \xi _{0}(x)=\pi ^{-1}\left(1+|x|^{2}\right)^{-1}}
7743:
The spherical transforms of smooth SO(2)-invariant functions of
5150:{\displaystyle \Delta =-\partial _{t}^{2}-\coth t\partial _{t}.}
4658:{\displaystyle g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
4172:{\displaystyle \xi _{0}(z)=\pi ^{-1}\left(1+|z|^{2}\right)^{-2}}
1981:{\displaystyle g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
26439:(2), American Journal of Mathematics, Vol. 80, No. 2: 241–310,
22479:(λ) can be attached to each such operator: namely the value at
15910:{\displaystyle {\overline {d(\lambda )}}{\tilde {F}}(\lambda )}
14447:. The corresponding character formula and dimension formula of
23286:{\displaystyle A(s,\lambda )\xi _{0}=c_{s}(\lambda )\xi _{0},}
18478:
is a normalising constant, given as a quotient of products of
6631:) = 1. There can only be one such solution either because the
2230:{\displaystyle ds^{2}=r^{-2}\left(dx^{2}+dy^{2}+dr^{2}\right)}
1215:{\displaystyle \lambda '(kx)=\Delta _{AN}(x)^{1/2}\lambda (x)}
209:
were naturally labelled by a parameter λ in the quotient of a
25748:. An equivalent computation that arises in the theory of the
14077:
of complex semisimple Lie algebras. The special case of SL(N,
6855:
If the spherical transform of an SO(2)-invariant function on
1724:{\displaystyle \|f\|^{2}=\int _{K}|f(k)|^{2}\,dk<\infty .}
52:
and has a similarly close interconnection with the theory of
23596:
is a real semisimple Lie group with real rank one, i.e. dim
23517:{\displaystyle \ell (s_{1}s_{2})=\ell (s_{1})+\ell (s_{2}).}
22335:
can be read off from this integral, leading to the formula:
20581:
can be expressed in terms of these operators by the formula
17173:{\displaystyle C_{c}^{\infty }(K_{0}\backslash G_{0}/K_{0})}
16789:{\displaystyle C_{c}^{\infty }(K_{0}\backslash G_{0}/K_{0})}
16643:
it follows that there is a canonical identification between
14665:, extend holomorphic to their complexifications. Moreover,
221:
in terms of this parametrization. Generalizing the ideas of
26888:
23029:{\displaystyle \ell (s_{1}s_{2})=\ell (s_{1})+\ell (s_{2})}
22451:. These operators exhibit the unitary equivalence between π
20507:
10950:
variable. A straightforward change of variables shows that
10859:
variable and therefore invariant under the transformations
7725:
3783:
3446:
968:
115:
22499:
is up to scalar multiplication the unique vector fixed by
19030:{\displaystyle m_{\alpha }=\dim {\mathfrak {g}}_{\alpha }}
11268:{\displaystyle \int _{H^{3}}F\,dV=\int _{H^{2}}M^{*}F\,dA}
9638:{\displaystyle \Delta _{n}=-L_{n}-R_{n}^{2}-(n-1)R_{n},\,}
6535:
around the rectangular indented contour with vertices at ±
5277:
orbital integrals (Harish-Chandra, Gelfand & Naimark).
1918:{\displaystyle {\mathfrak {H}}^{3}=\{x+yi+tj\mid t>0\}}
217:, it became a central problem to determine explicitly the
24328:
19996:{\displaystyle \alpha _{0}=(\alpha ,\alpha )^{-1}\alpha }
14627:{\displaystyle {\mathfrak {t}}^{*}\times {\mathfrak {t}}}
5274:
Abel's integral equation, following Selberg and Godement;
3462:
is unitary and sends the convolution operator defined by
26316:
The Plancherel Formula for Complex Semisimple Lie Groups
25182:{\displaystyle {\mathcal {S}}({\mathfrak {a}}^{*})^{W},}
5159:
The integral of an SO(2)-invariant function is given by
2547:
The integral of an SU(2)-invariant function is given by
26644:, Springer Monographs in Mathematics, Springer-Verlag,
22575:, the various choices being permuted by the Weyl group
21734:
Harish-Chandra obtained a second integral formula for φ
20800:{\displaystyle \Delta _{\mathfrak {a}}=-\sum X_{i}^{2}}
13886:). Combining this with the above inversion formula for
11732:{\displaystyle \Phi _{2\lambda }=M\varphi _{\lambda }.}
7828:{\displaystyle |F(\lambda )|\leq Ce^{R|\Im \lambda |}.}
7259:), this immediately implies the Plancherel formula for
6846:{\displaystyle \Phi _{\lambda }=M_{1}\phi _{\lambda }.}
4886:{\displaystyle ds^{2}=r^{-2}\left(dx^{2}+dr^{2}\right)}
4591:{\displaystyle {\mathfrak {H}}^{2}=\{x+ri\mid r>0\}}
26593:
Differential geometry, Lie groups and symmetric spaces
24950:
24045:
20288:
20152:
18485:
17139:
17069:
16828:
16755:
16590:
16531:
11541:
Thus, as in the case of Hadamard's method of descent.
10299:{\displaystyle \Delta _{3}Ef=E(\Delta _{2}-R_{2})f.\,}
8960:
8808:
8699:
8621:
8385:
7527:
6312:
6256:{\displaystyle f_{\lambda }=M_{1}^{*}\Phi _{\lambda }}
6162:
5654:
5494:
4624:
3491:
3354:
2732:{\displaystyle U^{*}\Delta U=-{d^{2} \over dt^{2}}+1.}
1947:
26092:"Sur les représentations induites des groupes de Lie"
25596:
25459:
25457:
25455:
25380:
25203:
25141:
24936:
24902:
24767:
24680:
24628:
24494:
24431:
24388:
24357:
24171:
24143:
by characterizing the spherical transforms of smooth
23980:
23658:
23621:-function can be computed directly by various means:
23560:
23437:
23319:
23219:
23152:
23107:
23071:
22952:
22834:
22732:
22613:
22550:
22341:
22306:
22229:
22202:
22032:
21929:
21862:
21831:
21756:
21702:
21606:
21575:
21484:
21467:
can be expressed in terms as a linear combination of
21442:
21380:
21295:
21184:
21138:
21038:
20946:
20849:
20813:
20760:
20706:
20682:
20587:
20486:
20462:
20426:
20382:
20321:
20265:
20102:
20024:
19951:
19615:
19467:
19366:
19315:
19264:
19198:
19166:
19135:
19111:
19087:
19051:
18993:
18946:
18831:
18798:
18774:
18750:
18671:
18628:
18583:
18559:
18528:
18500:
18456:
18318:
17941:
17747:
17632:
17521:
17470:
17398:
17280:
17186:
17110:
17047:
16996:
16861:
16806:
16726:
16579:
16527:
16503:
16454:
16214:
16086:
16028:
15927:
15864:
15721:
15697:
15653:
15390:
15294:
15164:
15118:
15074:
14923:
14878:
14824:
14673:
14640:
14599:
14459:
14422:
14365:
14289:
14260:
14198:
14174:
14150:
14102:
13901:
13784:
13700:
13644:
13565:
13253:
13134:
13099:
12797:
12671:
12464:
12363:
12180:
12088:
12009:
11853:
11747:
11696:
11614:
11549:
11479:
11367:
11285:
11197:
11127:
10956:
10886:
10814:{\displaystyle \Delta _{3}M_{0}f=4M_{0}\Delta _{2}f.}
10755:
10727:{\displaystyle (\Delta _{2}-R_{2})Sf=4S\Delta _{2}f,}
10663:
10518:
10332:
10237:
10164:
9943:
9770:
9655:
9558:
9517:
9511:
is related to hyperbolic distance from the origin by
9328:
9177:
8935:
8885:
8796:
8680:
8636:
8573:
8455:
8290:
8194:
8092:
7962:
7929:
7847:
7767:
7587:
7484:
7282:
7227:
7081:
6968:
6894:
6861:
6806:
6651:
6555:
6494:
6359:
6269:
6214:
6114:
5975:
5824:
5700:
5601:
5540:
5438:
5341:
5308:
5165:
5097:
5043:
4963:
4903:
4814:
4766:
4675:
4612:
4540:
4532:) acts transitively on the Poincaré upper half plane
4471:
4353:
4289:
4189:
4087:
3957:
3924:
3829:
3649:
3566:
3524:
3468:
3323:
3162:
3083:
3011:
2914:
2843:
2765:
2757:
can be expressed in terms of the spherical functions
2671:
2553:
2482:
2428:
2314:
2247:
2142:
2085:
1994:
1935:
1858:
1751:
1646:
1574:
1553:{\displaystyle f(gb)=\Delta (b)^{1/2}\lambda (b)f(g)}
1480:
1415:
can be identified with the matrix coefficient of the
1292:
1139:
977:
923:
794:
751:
664:
579:
536:
493:
461:
426:
25328:
25326:
24928:
The Harish-Chandra Schwartz space can be defined as
22544:, however, depends on a choice of a Weyl chamber in
22287:{\displaystyle \lim _{t\to \infty }e^{tX}ne^{-tX}=1}
20831:, corresponding to any choice of orthonormal basis (
19258:. A choice of positive roots defines a Weyl chamber
15149:{\displaystyle {\mathfrak {a}}=i{\mathfrak {t}}^{*}}
14408:
The finite-dimensional irreducible representations π
26215:
25343:
25314:
25312:
25310:
23547:
18656:{\displaystyle {\mathfrak {p}}={\mathfrak {g}}_{-}}
18611:{\displaystyle {\mathfrak {k}}={\mathfrak {g}}_{+}}
11532:{\displaystyle M^{*}\Delta _{3}=4\Delta _{2}M^{*}.}
11354:{\displaystyle M^{*}((Mf)\cdot F)=f\cdot (M^{*}F),}
5583:on S0(2)-invariant functions obtained by averaging
307:implicitly invokes the spherical transform; it was
27218:Spectral theory of ordinary differential equations
26570:
26477:(3), The Johns Hopkins University Press: 553–613,
26291:
25640:{\displaystyle \chi _{\lambda }d(\lambda )^{-1/2}}
25639:
25590:is equivalent to the statement that the functions
25452:
25390:
25302:
25223:
25181:
25125:
24912:
24878:
24754:{\displaystyle f_{1}(g)=\int _{K}f(x^{-1}kg)\,dk,}
24753:
24664:
24598:
24453:
24417:
24374:
24285:
24104:
23966:
23580:
23516:
23423:
23285:
23213:-functions are in general defined by the equation
23199:
23132:
23093:
23028:
22938:
22818:
22718:
22567:
22414:
22323:
22286:
22212:
22184:
22012:
21882:
21848:
21793:
21719:
21684:
21592:
21547:
21452:
21397:
21367:leads to a recursive formula for the coefficients
21358:{\displaystyle \coth x-1=2\sum _{m>0}e^{-2mx},}
21357:
21270:
21148:
21116:
21006:
20932:
20823:
20799:
20746:
20692:
20661:
20568:
20472:
20436:
20416:, it defines a second order differential operator
20392:
20336:
20307:
20251:
20082:
19995:
19937:
19594:
19447:
19332:
19286:
19250:
19176:
19152:
19121:
19097:
19061:
19029:
18975:
18930:
18815:
18784:
18760:
18734:
18655:
18610:
18569:
18545:
18510:
18466:
18436:
18278:
17927:
17700:
17616:
17496:
17443:
17382:
17256:
17172:
17088:
17027:
16924:
16847:
16788:
16633:
16563:
16513:
16482:
16324:
16188:
16050:
15968:
15909:
15850:
15707:
15675:
15637:
15372:
15245:
15148:
15097:
15058:
14888:
14860:
14801:
14657:
14626:
14583:
14439:
14398:
14343:
14273:
14234:
14184:
14160:
14112:
14085:; this group is also the normal real form of SL(N,
14045:
13864:
13766:
13661:
13628:
13535:
13235:
13114:
13079:
12779:
12641:
12446:
12341:
12164:
12074:
11984:
11817:
11731:
11682:
11600:
11531:
11463:
11353:
11267:
11181:
11072:
10938:
10813:
10726:
10631:
10502:
10298:
10219:
10126:
9905:
9752:
9637:
9535:
9473:
9235:
9158:
8919:
8871:
8768:
8653:
8606:
8557:
8439:
8371:
8270:
8176:
8074:
7946:
7896:
7827:
7733:
7558:
7458:
7244:
7211:
7040:
6950:
6878:
6845:
6788:
6613:
6527:
6488:since the integral can be computed by integrating
6478:
6343:
6255:
6198:
6092:
5944:
5806:
5675:
5557:
5518:
5420:
5325:
5258:spectral theory of ordinary differential equations
5242:
5149:
5060:
5027:
4942:
4885:
4783:
4742:
4657:
4590:
4504:
4455:
4322:
4265:
4171:
4069:
3941:
3892:
3791:
3620:
3539:
3510:
3452:
3301:
3139:
3048:
2995:
2896:
2825:
2731:
2629:
2537:
2445:
2412:
2293:
2229:
2119:
2061:
1980:
1917:
1806:
1723:
1632:
1552:
1347:
1214:
1054:
933:
882:
761:
674:
589:
553:
503:
471:
436:
227:spectral theory of ordinary differential equations
26287:, vol. 4, Soc. Math. France, pp. 95–110
25323:
24888:A further scaling argument allows the inequality
24488:This part of the Paley-Wiener theorem shows that
24038:
23863:
23775:
23554:corresponding to the Lie subalgebra generated by
22607:, λ) is defined on the induced representation by
22512:(λ). These operators all act on the same space L(
19837:
19749:
19302:consists of roots α such that α/2 is not a root.
18976:{\displaystyle {\mathfrak {g}}_{\alpha }\neq (0)}
17515:, the constant of proportionality being given by
17257:{\displaystyle M^{*}F(a^{2})=\int _{K}F(ga)\,dg.}
16634:{\displaystyle K_{0}\backslash G_{0}/K_{0}=A_{+}}
16497:is the image of the closure of a Weyl chamber in
16022:(λ) indexed by lattice points in the interior of
13688:)). On the other hand, for biinvariant functions
13545:This gives the spherical inversion for the point
13009:
12912:
12824:
12733:
12507:
12400:
12241:
7897:{\displaystyle (M_{1}^{*}F)^{\sim }={\tilde {F}}}
5421:{\displaystyle M_{1}f(x,y,r)=r^{1/2}\cdot f(x,r)}
1807:{\displaystyle \chi _{\lambda }(g)=(\pi (g)1,1).}
1439:, this is defined as the unitary representation π
1263:give the same spherical function if and only if λ
27344:
26768:Proceedings of the American Mathematical Society
25307:
25135:Under the spherical transform it is mapped onto
25019:
23200:{\displaystyle c(\lambda )=c_{s_{0}}(\lambda ).}
22231:
17089:{\displaystyle C_{c}^{\infty }(U\backslash G/U)}
16848:{\displaystyle C_{c}^{\infty }(U\backslash G/U)}
15156:and given by the Harish-Chandra-Berezin formula
15098:{\displaystyle {\mathfrak {a}}=i{\mathfrak {t}}}
14127:
13893:yields the general spherical inversion formula:
10939:{\displaystyle QF=\int _{K_{1}}F\circ g_{s}\,ds}
7612:
3674:
3140:{\displaystyle f^{*}(g)={\overline {f(g^{-1})}}}
1458:given by the composition of the homomorphism of
1073:the integral is with respect to Haar measure on
26499:"Discrete series for semisimple Lie groups, II"
26466:
26428:
26175:
26155:
25672:
25660:
25563:
25463:
25332:
25318:
24322:
18301:
17444:{\displaystyle M^{*}\Delta _{c}=4\Delta M^{*}.}
16445:
14093:
7579:belonging to the Harish-Chandra Schwartz space
5285:
3641:belonging to the Harish-Chandra Schwartz space
3311:For biinvariant functions this establishes the
281:
256:one, thus in particular covering the case when
178:gave a more concrete and satisfactory form for
26563:, Battelle Rencontres, Benjamin, pp. 1–71
26496:
26380:
26321:
26193:
25354:
25291:
25280:
25269:
22424:
18446:where α ranges over the positive roots of the
18309:
26874:
26635:
26612:Helgason, Sigurdur (2000), "Harish-Chandra's
26096:Bulletin de la Société Mathématique de France
25485:
24465:> 0. Using Harish-Chandra's expansion of φ
23581:{\displaystyle {\mathfrak {g}}_{\pm \alpha }}
22532:is uniquely determined as the centraliser of
20348:Harish-Chandra's spherical function expansion
16925:{\displaystyle Mf(a)=\int _{U}f(ua^{2})\,du.}
16426:be the corresponding Cartan decomposition of
16337:
14136:is a complex semisimple Lie group, it is the
14082:
13629:{\displaystyle f_{1}(w)=\int _{K}f(gkw)\,dk,}
8447:on the first factor, is given by the formula
7954:. The representation is given by the formula
3949:. The representation is given by the formula
1400:
21565:. It describes the asymptotic behaviour of φ
18922:
18849:
4585:
4558:
3518:into the multiplication operator defined by
1912:
1876:
1654:
1647:
734:-biinvariant square integrable functions on
561:can be identified with the commutant of the
26737:Scattering theory for automorphic functions
26731:
26618:Proceedings of Symposia in Pure Mathematics
26178:Analysis on non-Riemannian symmetric spaces
25435:
25374:under convolution, a dense *-subalgebra of
24147:-bivariant functions of compact support on
20308:{\displaystyle f\in L^{1}(K\backslash G/K)}
18288:The direct calculation of the integral for
17722:, then the spherical inversion formula for
11994:
7759:satisfying an exponential growth condition
3821:satisfying an exponential growth condition
3511:{\displaystyle f\in L^{1}(K\backslash G/K)}
229:, Harish-Chandra introduced his celebrated
26881:
26867:
24418:{\displaystyle {\mathfrak {a}}^{*}+i\mu t}
24314:This was proved by Helgason and Gangolli (
24299:is supported in the closed ball of radius
20747:{\displaystyle (A_{\alpha },X)=\alpha (X)}
20400:, that are invariant under the Weyl group
20094:Plancherel theorem for spherical functions
14073:Lie groups for which the Lie algebras are
9320:the volume and area elements are given by
8607:{\displaystyle L^{2}({\mathfrak {H}}^{2})}
8567:is unitary and gives the decomposition of
7575:are the spherical transforms of functions
4505:{\displaystyle L^{2}({\mathfrak {H}}^{3})}
4465:is unitary and gives the decomposition of
4323:{\displaystyle L^{2}({\mathfrak {H}}^{3})}
3637:are the spherical transforms of functions
3313:Plancherel theorem for spherical functions
1387:can be identified with the quotient space
22:Plancherel theorem for spherical functions
26850:
26836:
26828:
26780:
26761:
26702:
26550:
26514:
26412:
26394:
26364:
26346:
26134:
26053:
25989:
25520:
24866:
24741:
24589:
24333:
24139:The Paley-Wiener theorem generalizes the
22706:
22483:of the intertwining operator applied to ξ
22402:
22172:
22000:
20895:
20645:
20635:
20404:. In particular since the Laplacian Δ on
20259:is unitary and transforms convolution by
20239:
20132:
20131:
20130:
19582:
19549:
19435:
18902:
18901:
18703:
18702:
18266:
18228:
18197:
18196:
18120:
18082:
17915:
17849:
17830:
17829:
17604:
17244:
16912:
16179:
15628:
15624:
15522:
15518:
15363:
15046:
15009:
14946:
14896:. There is a similar product formula for
14060:All complex semisimple Lie groups or the
14033:
13616:
13523:
13408:
13330:
13223:
13046:
12968:
12961:
12858:
12767:
12629:
12567:
12434:
12329:
12322:
12152:
12062:
11972:
11927:
11451:
11403:
11258:
11218:
11060:
10977:
10929:
10295:
10216:
10114:
10059:
10038:
10037:
10036:
10026:
9964:
9902:
9840:
9839:
9838:
9707:
9706:
9705:
9634:
9461:
9454:
9404:
9403:
9402:
9392:
9385:
9378:
9236:{\displaystyle a^{2}=1+b^{2}+x^{2}+y^{2}}
8782:)/SU(2) can be identified with the space
8548:
8430:
8362:
7446:
7331:
7199:
6938:
6776:
6432:
6083:
6022:
5932:
5871:
5797:
5172:
4933:
4926:
4446:
3405:
3388:
3289:
3278:
3200:
2983:
2887:
2617:
2600:
2560:
2284:
2277:
2270:
1705:
1042:
870:
554:{\displaystyle {\mathfrak {A}}^{\prime }}
27171:Group algebra of a locally compact group
26611:
26590:
26568:
26558:
26531:
26275:
26250:Functional Analysis and Its Applications
26247:
26061:
26024:
26000:
25977:
25968:instead of the results of Mustapha Rais.
25952:
25940:
25928:
25916:
25868:
25856:
25844:
25832:
25820:
25808:
25796:
25784:
25773:
25761:
25753:
25684:
25575:
25551:
25508:
25496:
25446:
25257:
24315:
23133:{\displaystyle -{\mathfrak {a}}_{+}^{*}}
22444:
18293:
2120:{\displaystyle {\mathfrak {H}}^{3}=G/K.}
308:
126:
26796:
26678:
26565:(a general introduction for physicists)
24454:{\displaystyle {\mathfrak {a}}_{+}^{*}}
24134:
23094:{\displaystyle {\mathfrak {a}}_{+}^{*}}
21794:{\displaystyle G=\bigcup _{s\in W}BsB,}
19287:{\displaystyle {\mathfrak {a}}_{+}^{*}}
17028:{\displaystyle 4M\Delta =\Delta _{c}M.}
16051:{\displaystyle {\mathfrak {a}}_{+}^{*}}
16014:itself, are the normalized characters χ
15676:{\displaystyle {\mathfrak {a}}_{+}^{*}}
15382:and the spherical inversion formula by
13638:another rotation invariant function on
12456:the spherical transform can be written
11191:The integral formula above then yields
6623:satisfies the normalisation condition φ
4743:{\displaystyle g(w)=(aw+b)(cw+d)^{-1}.}
2062:{\displaystyle g(w)=(aw+b)(cw+d)^{-1}.}
1817:
1633:{\displaystyle \pi (g)f(x)=f(g^{-1}x),}
315:) along the lines sketched by Selberg.
304:
27345:
26113:
26089:
25424:
24329:Rosenberg's proof of inversion formula
22528:has been chosen, the compact subgroup
22448:
21811:and the union is disjoint. Taking the
21420:is also an eigenfunction of the other
21032:is an eigenfunction of the Laplacian:
18546:{\displaystyle {\mathfrak {g}}_{\pm }}
17497:{\displaystyle M^{*}\Phi _{2\lambda }}
16564:{\displaystyle K\backslash G/U=A_{+}.}
15969:{\displaystyle F(e^{X})\delta (e^{X})}
15687:. In fact the result follows from the
14055:
6528:{\displaystyle e^{i\lambda t}/\sinh t}
5593:by the action of SU(2) also satisfies
3049:{\displaystyle f=f_{2}^{*}\star f_{1}}
1737:can be identified with functions in L(
1096:is the Abelian vector subgroup in the
917:) denotes the convolution operator in
318:
26862:
26816:Rocky Mountain Journal of Mathematics
26812:
26657:
26039:
26012:
25904:
25892:
25880:
25757:
25474:
25234:
24923:
21424:-invariant differential operators on
18292:(λ), generalising the computation of
16934:Here a third Cartan decomposition of
16521:under the exponential map. Moreover,
14593:These formulas, initially defined on
14144:, a compact semisimple Lie group. If
2663:, Δ is transformed into the operator
2294:{\displaystyle dV=r^{-3}\,dx\,dy\,dr}
378:-biinvariant continuous functions on
374:), consisting of compactly supported
26752:
26710:
25699:(λ) can be written as integral over
25586:The spherical inversion formula for
25531:
25224:{\displaystyle {\mathfrak {a}}^{*}.}
21883:{\displaystyle -{\mathfrak {a}}_{+}}
21436:-invariant differential operator on
20372:can be identified with functions on
16697:can be identified with functions on
16353:of the complex semisimple Lie group
10312:is SO(2)-invariant, then, regarding
4519:
2906:and the spherical inversion formula
1835:
1348:{\displaystyle W=N_{K}(A)/C_{K}(A),}
1129:, using the group homomorphism onto
941:and the integral is with respect to
658:, μ) which carries the operators in
36:. It is a natural generalisation in
27353:Representation theory of Lie groups
25383:
25207:
25155:
24905:
24785:
24518:
24435:
24392:
24375:{\displaystyle {\mathfrak {a}}^{*}}
24361:
23652:This yields the following formula:
23564:
23114:
23075:
22568:{\displaystyle {\mathfrak {a}}^{*}}
22554:
22435:The many roles of Harish-Chandra's
22324:{\displaystyle {\mathfrak {a}}_{+}}
22310:
22205:
21869:
21849:{\displaystyle {\mathfrak {a}}_{+}}
21835:
21720:{\displaystyle {\mathfrak {a}}_{+}}
21706:
21593:{\displaystyle {\mathfrak {a}}_{+}}
21579:
21445:
21398:{\displaystyle {\mathfrak {a}}_{+}}
21384:
21141:
20966:
20816:
20767:
20685:
20600:
20465:
20429:
20385:
20188:
19491:
19333:{\displaystyle {\mathfrak {a}}^{*}}
19319:
19268:
19251:{\displaystyle W=N_{K}(A)/C_{K}(A)}
19169:
19153:{\displaystyle {\mathfrak {a}}^{*}}
19139:
19114:
19090:
19054:
19016:
18950:
18914:
18860:
18835:
18816:{\displaystyle {\mathfrak {a}}^{*}}
18802:
18777:
18768:be a maximal Abelian subalgebra of
18753:
18718:
18694:
18684:
18674:
18642:
18631:
18597:
18586:
18562:
18532:
18503:
18486:Harish-Chandra's Plancherel theorem
18459:
18138:
17990:
17771:
16506:
16394:. The fixed point subgroup of τ is
16032:
15700:
15657:
15537:
15436:
15135:
15121:
15090:
15077:
14984:
14881:
14658:{\displaystyle {\mathfrak {t}}^{*}}
14644:
14619:
14603:
14535:
14440:{\displaystyle {\mathfrak {t}}^{*}}
14426:
14333:
14307:
14263:
14224:
14211:
14201:
14177:
14153:
14105:
13662:{\displaystyle {\mathfrak {H}}^{2}}
13648:
10220:{\displaystyle Ef(b,x,y)=f(b,x).\,}
9923:and an SO(2)-invariant function on
8654:{\displaystyle {\mathfrak {H}}^{3}}
8640:
8622:Flensted–Jensen's method of descent
8618:of the spherical principal series.
8590:
8313:
7947:{\displaystyle {\mathfrak {H}}^{2}}
7933:
7923:is identified with the boundary of
7291:
7245:{\displaystyle {\mathfrak {H}}^{2}}
7231:
6879:{\displaystyle {\mathfrak {H}}^{2}}
6865:
6040:
5984:
5889:
5833:
5558:{\displaystyle {\mathfrak {H}}^{n}}
5544:
5326:{\displaystyle {\mathfrak {H}}^{2}}
5312:
5061:{\displaystyle {\mathfrak {H}}^{2}}
5047:
4784:{\displaystyle {\mathfrak {H}}^{2}}
4770:
4544:
4516:of the spherical principal series.
4488:
4306:
3942:{\displaystyle {\mathfrak {H}}^{3}}
3928:
3918:is identified with the boundary of
2446:{\displaystyle {\mathfrak {H}}^{3}}
2432:
2089:
1862:
926:
754:
667:
582:
540:
496:
464:
429:
106:, these expansions were known from
13:
26616:-function: a mathematical jewel",
25144:
25112:
25073:
24986:
24939:
24477:, the integral can be bounded for
24035:
23860:
23772:
23736:
22241:
21228:
21178:in terms of a perturbation series
21039:
20961:
20762:
20595:
20053:
19834:
19746:
19710:
19656:
19305:Defining the spherical functions φ
18304:. An explicit product formula for
18300:), was left as an open problem by
18223:
18220:
18217:
18077:
18074:
18071:
18043:
17844:
17841:
17838:
17583:
17548:
17482:
17425:
17410:
17121:
17058:
17010:
17003:
16817:
16737:
16361:. Let τ be a Cartan involution of
15809:
15806:
15803:
15800:
15754:
15577:
15471:
15342:
15259:induced from the character of the
15166:
14733:
14730:
14727:
14724:
13944:
13453:
13448:
13350:
13345:
13301:
13176:
12996:
12899:
12894:
12881:
12876:
12811:
12806:
12720:
12715:
12587:
12499:
12392:
12387:
12233:
12218:
12213:
11882:
11877:
11698:
11641:
11561:
11507:
11491:
11182:{\displaystyle M^{*}F(t)=QF(t/2).}
10796:
10757:
10709:
10668:
10617:
10578:
10565:
10520:
10485:
10437:
10421:
10402:
10372:
10340:
10264:
10239:
10088:
10083:
9993:
9988:
9890:
9874:
9858:
9821:
9803:
9785:
9741:
9725:
9688:
9670:
9560:
8352:
8196:
7809:
7716:
7664:
7590:
7486:
7354:
7349:
7110:
7105:
6808:
6602:
6271:
6244:
6149:
6131:
5768:
5763:
5641:
5603:
5481:
5440:
5195:
5190:
5135:
5108:
5098:
5005:
4987:
4964:
4191:
3774:
3726:
3652:
3606:
3571:
3567:
2955:
2875:
2767:
2682:
2583:
2578:
2523:
2493:
2483:
2398:
2374:
2356:
2338:
2315:
1715:
1499:
1164:
1015:
546:
14:
27374:
26782:10.1090/S0002-9939-1977-0507231-8
26303:, American Mathematical Society,
24382:; the integral can be shifted to
23629:can be expressed in terms of the
23548:Gindikin & Karpelevich (1962)
22441:non-commutative harmonic analysis
16437:be a maximal Abelian subgroup of
11101:is SU(2)-invariant. In this case
9280:of real-valued positive matrices
8920:{\displaystyle g\cdot A=gAg^{*}.}
6263:is SO(2)-invariant and satisfies
4943:{\displaystyle dA=r^{-2}\,dx\,dr}
627:, there is an essentially unique
38:non-commutative harmonic analysis
27327:
27326:
27253:Topological quantum field theory
26176:Flensted-Jensen, Mogens (1986),
26119:Heat Kernels and Spectral Theory
24321:The theorem was later proved by
24155:-invariant and that there is an
23526:This reduces the computation of
22487:, the constant function 1, in L(
17274:* satisfy the duality relations
14274:{\displaystyle {\mathfrak {t}}.}
14140:of its maximal compact subgroup
9915:For an SU(2)-invariant function
4756:is the maximal compact subgroup
4345:λ on the first factor) given by
2643:) by the unitary transformation
2075:is the maximal compact subgroup
27363:Theorems in functional analysis
26704:10.1090/S0002-9904-1969-12235-4
26561:Lie groups and symmetric spaces
26470:American Journal of Mathematics
26432:American Journal of Mathematics
26018:
26006:
25994:
25983:
25971:
25958:
25946:
25934:
25922:
25910:
25898:
25886:
25874:
25862:
25850:
25838:
25826:
25814:
25802:
25790:
25778:
25767:
25729:is the Cartan decomposition of
25690:
25678:
25666:
25654:
25580:
25569:
25557:
25545:
25536:
25525:
25514:
25502:
25490:
25479:
25468:
25440:
25429:
25418:
25401:
25391:{\displaystyle {\mathfrak {A}}}
25360:
25344:Gindikin & Karpelevich 1962
24913:{\displaystyle {\mathfrak {a}}}
22331:, the asymptotic behaviour of φ
22213:{\displaystyle {\mathfrak {a}}}
21453:{\displaystyle {\mathfrak {a}}}
21413:. Remarkably it turns out that
21149:{\displaystyle {\mathfrak {a}}}
20824:{\displaystyle {\mathfrak {a}}}
20693:{\displaystyle {\mathfrak {a}}}
20473:{\displaystyle {\mathfrak {a}}}
20437:{\displaystyle {\mathfrak {a}}}
20393:{\displaystyle {\mathfrak {a}}}
19177:{\displaystyle {\mathfrak {a}}}
19122:{\displaystyle {\mathfrak {a}}}
19098:{\displaystyle {\mathfrak {p}}}
19062:{\displaystyle {\mathfrak {a}}}
18785:{\displaystyle {\mathfrak {p}}}
18761:{\displaystyle {\mathfrak {a}}}
18570:{\displaystyle {\mathfrak {g}}}
18511:{\displaystyle {\mathfrak {g}}}
18467:{\displaystyle {\mathfrak {a}}}
16514:{\displaystyle {\mathfrak {a}}}
15708:{\displaystyle {\mathfrak {a}}}
14889:{\displaystyle {\mathfrak {t}}}
14546:
14399:{\displaystyle G=P\cdot U=UAU.}
14315:
14185:{\displaystyle {\mathfrak {u}}}
14161:{\displaystyle {\mathfrak {g}}}
14113:{\displaystyle {\mathfrak {a}}}
12659:is classically inverted by the
11361:the following adjoint formula:
8879:with the group action given by
2753:, any SU(2)-invariant function
2239:with associated volume element
1407:Principal series representation
934:{\displaystyle {\mathfrak {A}}}
762:{\displaystyle {\mathfrak {A}}}
675:{\displaystyle {\mathfrak {A}}}
590:{\displaystyle {\mathfrak {A}}}
504:{\displaystyle {\mathfrak {A}}}
472:{\displaystyle {\mathfrak {A}}}
437:{\displaystyle {\mathfrak {A}}}
26665:, Princeton University Press,
26642:Spherical inversion on SL(n,R)
26121:, Cambridge University Press,
25617:
25610:
25348:
25337:
25303:Guillemin & Sternberg 1977
25296:
25285:
25274:
25263:
25251:
25167:
25149:
25101:
25095:
25083:
25070:
25061:
25057:
25045:
25033:
25006:
24991:
24964:
24944:
24853:
24848:
24842:
24835:
24821:
24815:
24809:
24738:
24716:
24697:
24691:
24656:
24650:
24638:
24632:
24576:
24571:
24565:
24558:
24554:
24548:
24542:
24504:
24498:
24239:
24234:
24226:
24216:
24199:
24195:
24189:
24183:
24173:
24141:classical Paley-Wiener theorem
24091:
24056:
23945:
23926:
23847:
23828:
23767:
23764:
23745:
23739:
23731:
23712:
23682:
23676:
23508:
23495:
23486:
23473:
23464:
23441:
23418:
23412:
23392:
23376:
23353:
23347:
23297:is the constant function 1 in
23267:
23261:
23235:
23223:
23191:
23185:
23162:
23156:
23023:
23010:
23001:
22988:
22979:
22956:
22930:
22911:
22905:
22876:
22867:
22838:
22810:
22798:
22792:
22786:
22767:
22761:
22748:
22736:
22703:
22691:
22661:
22655:
22641:
22635:
22629:
22617:
22601:standard intertwining operator
22393:
22387:
22371:
22365:
22351:
22345:
22238:
22166:
22137:
22119:
22113:
22095:
22089:
22056:
22043:
21988:
21978:
21946:
21940:
21671:
21656:
21648:
21642:
21633:
21617:
21526:
21517:
21249:
21243:
21089:
21083:
21068:
21062:
20914:
20896:
20741:
20735:
20726:
20707:
20543:
20528:
20499:
20493:
20328:
20302:
20282:
20246:
20226:
20221:
20215:
20208:
20182:
20169:
20166:
20146:
20121:
20112:
19978:
19965:
19919:
19900:
19821:
19802:
19741:
19738:
19719:
19713:
19705:
19686:
19625:
19619:
19569:
19564:
19558:
19551:
19546:
19540:
19527:
19521:
19515:
19477:
19471:
19432:
19426:
19410:
19404:
19385:
19379:
19373:
19245:
19239:
19221:
19215:
18970:
18964:
18919:
18903:
18895:
18889:
18880:
18868:
18663:give the Cartan decomposition
18553:be the ±1 eigenspaces of σ in
18425:
18413:
18408:
18396:
18353:
18343:
18328:
18322:
18256:
18251:
18242:
18235:
18207:
18201:
18193:
18187:
18174:
18168:
18162:
18110:
18105:
18096:
18089:
18061:
18055:
18029:
18020:
18014:
17976:
17970:
17951:
17945:
17905:
17900:
17891:
17884:
17876:
17872:
17866:
17859:
17826:
17820:
17807:
17801:
17795:
17757:
17751:
17692:
17683:
17677:
17665:
17659:
17650:
17633:
17601:
17595:
17566:
17560:
17531:
17525:
17374:
17358:
17308:
17299:
17241:
17232:
17213:
17200:
17167:
17126:
17083:
17063:
16909:
16893:
16874:
16868:
16842:
16822:
16783:
16742:
16708:that are left invariant under
16387:in a maximal compact subgroup
16310:
16303:
16294:
16288:
16280:
16274:
16258:
16252:
16246:
16224:
16218:
16173:
16167:
16154:
16148:
16130:
16124:
16105:
16099:
16093:
15963:
15950:
15944:
15931:
15904:
15898:
15892:
15877:
15871:
15840:
15834:
15820:
15814:
15776:
15763:
15750:
15737:
15731:
15725:
15614:
15609:
15603:
15596:
15592:
15586:
15573:
15567:
15561:
15508:
15503:
15497:
15490:
15486:
15480:
15467:
15461:
15455:
15421:
15413:
15400:
15394:
15360:
15354:
15338:
15332:
15313:
15307:
15301:
15234:
15228:
15220:
15207:
15188:
15175:
15036:
15031:
15018:
15011:
15006:
14993:
14971:
14963:
14943:
14937:
14847:
14841:
14790:
14777:
14767:
14758:
14744:
14738:
14697:
14684:
14556:
14550:
14540:
14524:
14518:
14505:
14483:
14470:
13986:
13980:
13967:
13961:
13955:
13911:
13905:
13856:
13850:
13834:
13828:
13822:
13810:
13804:
13792:
13751:
13745:
13739:
13717:
13711:
13613:
13601:
13582:
13576:
13476:
13470:
13464:
13324:
13318:
13312:
13263:
13257:
13220:
13211:
13202:
13193:
13187:
13150:
13138:
13106:
13071:
13065:
12681:
12675:
12607:
12595:
12483:
12477:
12471:
12373:
12367:
12199:
12193:
12187:
12149:
12143:
12132:
12126:
12107:
12101:
12095:
12059:
12050:
12026:
12020:
11924:
11918:
11912:
11903:
11897:
11863:
11857:
11809:
11800:
11794:
11782:
11776:
11766:
11750:
11659:
11653:
11624:
11618:
11585:
11579:
11448:
11436:
11394:
11385:
11345:
11329:
11317:
11308:
11299:
11296:
11173:
11159:
11147:
11141:
11037:
11004:
10690:
10664:
10613:
10604:
10561:
10537:
10481:
10457:
10362:
10333:
10286:
10260:
10210:
10198:
10189:
10171:
10102:
10096:
10007:
10001:
9618:
9606:
9434:
9414:
9358:
9338:
9101:
9065:
9049:
9013:
8601:
8584:
8545:
8539:
8526:
8520:
8507:
8501:
8474:
8462:
8366:
8355:
8343:
8340:
8327:
8324:
8307:
8262:
8236:
8230:
8217:
8211:
8205:
8150:
8141:
8109:
8103:
8066:
8063:
8057:
8051:
8026:
8008:
8001:
7995:
7989:
7973:
7888:
7870:
7848:
7816:
7805:
7786:
7782:
7776:
7769:
7705:
7699:
7686:
7680:
7668:
7655:
7646:
7626:
7392:
7366:
7152:
7146:
7140:
7131:
7125:
7091:
7085:
7032:
7026:
7020:
7008:
7002:
6992:
6971:
6913:
6907:
6901:
6678:
6662:
6579:
6572:
6391:
6385:
6369:
6363:
6104:and SO(2)-invariant functions
6100:for SU(2)-invariant functions
6080:
6059:
6013:
5997:
5929:
5908:
5862:
5846:
5794:
5776:
5731:
5719:
5415:
5403:
5373:
5355:
5209:
5203:
5019:
4983:
4725:
4709:
4706:
4691:
4685:
4679:
4499:
4482:
4443:
4437:
4424:
4418:
4405:
4399:
4372:
4360:
4317:
4300:
4257:
4231:
4225:
4212:
4206:
4200:
4145:
4136:
4104:
4098:
4061:
4058:
4052:
4046:
4021:
4003:
3996:
3990:
3984:
3968:
3848:
3844:
3838:
3831:
3763:
3757:
3748:
3742:
3730:
3717:
3708:
3688:
3602:
3583:
3531:
3505:
3485:
3437:
3428:
3412:
3384:
3371:
3368:
3348:
3269:
3263:
3251:
3238:
3232:
3220:
3128:
3112:
3100:
3094:
2970:
2964:
2951:
2945:
2939:
2924:
2918:
2862:
2856:
2850:
2782:
2776:
2597:
2591:
2388:
2334:
2044:
2028:
2025:
2010:
2004:
1998:
1798:
1786:
1780:
1774:
1768:
1762:
1695:
1690:
1684:
1677:
1624:
1605:
1596:
1590:
1584:
1578:
1547:
1541:
1535:
1529:
1509:
1502:
1493:
1484:
1339:
1333:
1315:
1309:
1209:
1203:
1183:
1176:
1157:
1148:
1030:
1020:
994:
988:
867:
861:
845:
839:
820:
817:
811:
805:
597:leaves invariant the subspace
24:is an important result in the
1:
27358:Theorems in harmonic analysis
27049:Uniform boundedness principle
26383:Proc. Natl. Acad. Sci. U.S.A.
26327:Proc. Natl. Acad. Sci. U.S.A.
26140:Treatise on Analysis, Vol. VI
26033:
25242:, using classical estimates.
21825:, the unique element mapping
19340:, the spherical transform of
16368:extended to an involution of
16066:. The spherical transform of
15278:The spherical transform of a
14192:are their Lie algebras, then
14128:Complex semisimple Lie groups
10823:The same relation holds with
9252:only changes the coordinates
4895:with associated area element
639:and a unitary transformation
382:, acts by convolution on the
81:direct integral decomposition
56:. It is the special case for
26552:10.1016/0001-8708(70)90037-X
26170:10.1016/0022-1236(78)90058-7
26055:10.1016/0022-1236(91)90065-D
24116:Gindikin–Karpelevich formula
22397:
22123:
21028:Now the spherical function φ
20450:radial part of the Laplacian
20412:commutes with the action of
20337:{\displaystyle {\tilde {f}}}
19081:defines an inner product on
18518:denote its Lie algebra. Let
17455:universal enveloping algebra
16158:
15994:the compact symmetric space
15881:
14861:{\displaystyle W=N_{U}(T)/T}
14416:are indexed by certain λ in
13115:{\displaystyle {\tilde {f}}}
7397:
7326:
5286:Hadamard's method of descent
4752:The stabiliser of the point
3540:{\displaystyle {\tilde {f}}}
3273:
3195:
3132:
2071:The stabiliser of the point
68:of radial functions for the
7:
26591:Helgason, Sigurdur (1978),
26569:Helgason, Sigurdur (1984),
26559:Helgason, Sigurdur (1968),
24674:By applying this result to
24665:{\displaystyle T(f)=Cf(o).}
24614:with support at the origin
23609:is just the Harish-Chandra
22431:Harish-Chandra's c-function
22425:Harish-Chandra's c-function
21563:Harish-Chandra's c-function
19604:Harish-Chandra's c-function
19459:spherical inversion formula
15108:The spherical functions of
14451:give explicit formulas for
14083:Jorgenson & Lang (2001)
10734:leading to the fundamental
9927:, regarded as functions of
7747:are precisely functions on
3809:are precisely functions on
2835:by the spherical transform
1848:) acts transitively on the
695:spherical Fourier transform
614:abelian von Neumann algebra
205:these spherical functions φ
85:irreducible representations
32:, due in its final form to
10:
27379:
27192:Invariant subspace problem
26735:; Phillips, Ralph (1976),
26314:, Appendix to Chapter VI,
26229:Doklady Akademii Nauk SSSR
26204:Doklady Akademii Nauk SSSR
25370:-biinvariant functions on
25292:Gelfand & Naimark 1948
24606:defines a distribution on
24159:> 0 such that for each
22428:
21432:, each of which induces a
16964:Let Δ be the Laplacian on
16942:has been used to identify
16712:and right invariant under
16690:-biinvariant functions on
16483:{\displaystyle G=KA_{+}U,}
16398:, the complexification of
16338:Real semisimple Lie groups
14081:) is treated in detail in
7751:which are restrictions of
6798:Similarly it follows that
3813:which are restrictions of
1417:spherical principal series
1404:
1401:Spherical principal series
1255:Two different characters λ
1118:by first extending λ to a
322:
132:
27322:
27281:
27205:
27184:
27143:
27082:
27024:
26970:
26912:
26905:
26262:10.1007/s10688-008-0042-2
26090:Bruhat, François (1956),
25486:Jorgenson & Lang 2001
25411:of μ in the sense of the
22540:. The nilpotent subgroup
21894:) has a dense open orbit
21474:and its transforms under
21163:and its transforms under
20368:that are invariant under
15689:Fourier inversion formula
13124:Fourier inversion formula
10868:. On the other hand, for
10838:is obtained by averaging
9550:are given by the formula
9536:{\displaystyle r=\sinh t}
8786:of positive 2×2 matrices
8778:The symmetric space SL(2,
6547:. Thus the eigenfunction
5568:Since the action of SL(2,
2747:Fourier inversion formula
952:The spherical functions φ
293:Poisson summation formula
58:zonal spherical functions
46:Fourier inversion formula
27161:Spectrum of a C*-algebra
26224:
26221:Karpelevich, Fridrikh I.
25651:for the class functions.
25245:
22495:). Equivalently, since ξ
22466:in the Weyl group and a
22022:can be transferred to σ(
19077:.The restriction of the
14900:(λ), a polynomial in λ.
11999:The spherical function φ
11995:Abel's integral equation
11847:then immediately yields
11843:) inversion formula for
7468:The spherical function φ
7069:) inversion formula for
5961:*, defined by averaging
3550:The spherical function Φ
1411:The spherical function φ
962:Harish-Chandra's formula
684:multiplication operators
346:maximal compact subgroup
325:Zonal spherical function
169:maximal compact subgroup
27258:Noncommutative geometry
26839:Bull. Soc. Math. France
26830:10.1216/rmjm/1181071659
26538:Advances in Mathematics
26497:Harish-Chandra (1966),
25735:Cartan-Helgason theorem
25436:Lax & Phillips 1976
23639:connection coefficients
23631:hypergeometric function
22591:' is the normaliser of
21914:initially defined over
21132:-invariant function on
20315:into multiplication by
19192:can be identified with
17626:Moreover, in this case
11093:is SO(2)--invariant if
10872:a suitable function on
10738:of Flensted-Jensen for
9292:). Taking coordinates (
5262:hypergeometric equation
1373:finite reflection group
714:can be identified with
682:onto the corresponding
623:is maximal Abelian. By
289:Selberg's trace formula
215:finite reflection group
155:, so in particular the
66:eigenfunction expansion
27314:Tomita–Takesaki theory
27289:Approximation property
27233:Calculus of variations
26753:Loeb, Jacques (1979),
26691:Bull. Amer. Math. Soc.
26630:10.1090/pspum/068/0834
26348:10.1073/pnas.37.12.813
25752:has been discussed by
25641:
25392:
25225:
25183:
25127:
24914:
24880:
24755:
24666:
24600:
24455:
24419:
24376:
24323:Flensted-Jensen (1986)
24287:
24106:
23968:
23582:
23518:
23425:
23287:
23201:
23134:
23095:
23030:
22940:
22820:
22720:
22569:
22416:
22325:
22288:
22214:
22186:
22014:
21884:
21850:
21795:
21721:
21686:
21594:
21549:
21454:
21399:
21359:
21272:
21167:are eigenfunctions of
21150:
21118:
21008:
20934:
20825:
20801:
20748:
20694:
20663:
20570:
20474:
20438:
20394:
20338:
20309:
20253:
20084:
19997:
19939:
19596:
19449:
19334:
19288:
19252:
19178:
19160:to be identified with
19154:
19123:
19099:
19063:
19031:
18977:
18932:
18817:
18786:
18762:
18736:
18657:
18618:is the Lie algebra of
18612:
18571:
18547:
18512:
18468:
18438:
18302:Flensted-Jensen (1978)
18280:
17929:
17702:
17618:
17498:
17445:
17384:
17258:
17174:
17090:
17029:
16926:
16849:
16790:
16635:
16565:
16515:
16484:
16446:Flensted-Jensen (1978)
16326:
16190:
16052:
15970:
15911:
15852:
15709:
15677:
15639:
15374:
15282:-biinvariant function
15247:
15150:
15099:
15060:
14911:-biinvariant function
14890:
14862:
14803:
14659:
14628:
14585:
14441:
14400:
14345:
14275:
14236:
14186:
14162:
14114:
14094:Flensted-Jensen (1978)
14047:
13866:
13768:
13663:
13630:
13537:
13237:
13116:
13081:
12781:
12661:Abel integral equation
12643:
12448:
12343:
12166:
12076:
11986:
11819:
11733:
11684:
11602:
11533:
11465:
11355:
11269:
11183:
11074:
10946:is independent of the
10940:
10815:
10728:
10633:
10504:
10300:
10221:
10128:
9907:
9754:
9639:
9537:
9475:
9237:
9160:
8921:
8873:
8770:
8655:
8608:
8559:
8441:
8373:
8272:
8186:is fixed by SO(2) and
8178:
8076:
7948:
7898:
7829:
7735:
7560:
7460:
7246:
7213:
7042:
6952:
6880:
6847:
6790:
6615:
6529:
6480:
6345:
6257:
6200:
6094:
5969:over SO(2), satisfies
5946:
5808:
5677:
5559:
5520:
5422:
5327:
5244:
5151:
5062:
5029:
4944:
4887:
4785:
4744:
4659:
4602:Möbius transformations
4592:
4506:
4457:
4324:
4267:
4181:is fixed by SU(2) and
4173:
4071:
3943:
3894:
3793:
3622:
3541:
3512:
3454:
3303:
3141:
3050:
2997:
2898:
2827:
2733:
2631:
2539:
2447:
2414:
2295:
2231:
2121:
2063:
1982:
1927:Möbius transformations
1919:
1808:
1725:
1634:
1554:
1349:
1216:
1056:
935:
884:
773:) can be described by
763:
697:or sometimes just the
676:
608:-invariant vectors in
591:
563:regular representation
555:
522:of a set of operators
505:
473:
438:
282:Flensted-Jensen (1978)
89:regular representation
54:differential equations
27309:Banach–Mazur distance
27272:Generalized functions
26396:10.1073/pnas.38.4.337
26301:Geometric Asymptotics
25642:
25413:Radon–Nikodym theorem
25393:
25226:
25184:
25128:
24915:
24881:
24756:
24667:
24601:
24469:and the formulas for
24456:
24420:
24377:
24288:
24163:there is an estimate
24107:
23969:
23583:
23519:
23426:
23288:
23202:
23135:
23096:
23031:
22941:
22821:
22721:
22570:
22417:
22326:
22289:
22215:
22187:
22015:
21885:
21851:
21796:
21722:
21687:
21595:
21550:
21455:
21400:
21360:
21273:
21151:
21119:
21009:
20935:
20826:
20802:
20749:
20695:
20664:
20571:
20475:
20439:
20395:
20339:
20310:
20254:
20085:
19998:
19940:
19597:
19450:
19335:
19289:
19253:
19179:
19155:
19124:
19100:
19064:
19032:
18983:, then α is called a
18978:
18933:
18818:
18787:
18763:
18737:
18658:
18613:
18572:
18548:
18513:
18469:
18439:
18310:Harish-Chandra (1966)
18281:
17930:
17703:
17619:
17499:
17446:
17385:
17259:
17175:
17091:
17030:
16927:
16850:
16791:
16636:
16566:
16516:
16485:
16327:
16191:
16053:
15971:
15912:
15853:
15710:
15678:
15640:
15375:
15248:
15151:
15112:are labelled by λ in
15100:
15061:
14903:On the complex group
14891:
14863:
14804:
14660:
14629:
14586:
14442:
14401:
14346:
14276:
14237:
14187:
14163:
14115:
14048:
13867:
13769:
13664:
13631:
13538:
13238:
13117:
13089:The relation between
13082:
12782:
12651:The relation between
12644:
12449:
12344:
12167:
12077:
11987:
11820:
11734:
11685:
11603:
11534:
11466:
11356:
11275:and hence, since for
11270:
11184:
11097:is, in particular if
11089:commutes with SO(2),
11075:
10941:
10816:
10729:
10634:
10505:
10301:
10222:
10129:
9908:
9755:
9640:
9538:
9476:
9276:unchanged. The space
9238:
9161:
8922:
8874:
8771:
8656:
8609:
8560:
8442:
8374:
8280:The representations π
8273:
8179:
8077:
7949:
7899:
7830:
7753:holomorphic functions
7736:
7561:
7461:
7247:
7214:
7043:
6953:
6881:
6848:
6791:
6616:
6530:
6481:
6346:
6258:
6201:
6095:
5947:
5809:
5685:The adjoint operator
5678:
5560:
5521:
5423:
5328:
5264:(Mehler, Weyl, Fock);
5245:
5152:
5063:
5030:
4945:
4888:
4786:
4745:
4660:
4593:
4507:
4458:
4325:
4275:The representations π
4268:
4174:
4072:
3944:
3895:
3815:holomorphic functions
3794:
3623:
3542:
3513:
3455:
3304:
3142:
3051:
2998:
2899:
2828:
2734:
2632:
2540:
2448:
2415:
2296:
2232:
2122:
2064:
1983:
1929:. The complex matrix
1920:
1809:
1726:
1635:
1555:
1350:
1217:
1098:Iwasawa decomposition
1057:
936:
885:
764:
677:
592:
556:
506:
474:
439:
153:semisimple Lie groups
142:locally compact group
30:semisimple Lie groups
26:representation theory
27054:Kakutani fixed-point
27039:Riesz representation
26802:J. Indian Math. Soc.
25673:Flensted-Jensen 1978
25661:Flensted-Jensen 1978
25594:
25564:Flensted-Jensen 1978
25464:Flensted-Jensen 1978
25378:
25333:Harish-Chandra 1958b
25319:Harish-Chandra 1958a
25201:
25139:
24934:
24900:
24765:
24678:
24626:
24492:
24429:
24386:
24355:
24303:about the origin in
24169:
24135:Paley–Wiener theorem
23978:
23656:
23558:
23435:
23317:
23217:
23150:
23105:
23069:
22950:
22832:
22730:
22611:
22548:
22339:
22304:
22227:
22200:
22030:
21927:
21890:, it follows that σ(
21860:
21829:
21754:
21744:Bruhat decomposition
21700:
21604:
21573:
21482:
21440:
21378:
21293:
21182:
21136:
21036:
20944:
20847:
20811:
20807:is the Laplacian on
20758:
20704:
20680:
20585:
20484:
20460:
20424:
20380:
20319:
20263:
20100:
20096:states that the map
20022:
19949:
19613:
19465:
19364:
19313:
19262:
19196:
19164:
19133:
19109:
19085:
19049:
18991:
18944:
18829:
18796:
18772:
18748:
18669:
18626:
18581:
18557:
18526:
18498:
18454:
18316:
17939:
17745:
17630:
17519:
17504:is proportional to φ
17468:
17396:
17278:
17184:
17108:
17045:
16994:
16982:be the Laplacian on
16859:
16804:
16724:
16704:as can functions on
16577:
16525:
16501:
16452:
16212:
16084:
16026:
15925:
15862:
15719:
15695:
15651:
15388:
15292:
15162:
15116:
15072:
14921:
14915:can be evaluated as
14907:, the integral of a
14876:
14822:
14671:
14638:
14597:
14457:
14420:
14363:
14355:Cartan decomposition
14287:
14258:
14196:
14172:
14148:
14100:
13899:
13782:
13698:
13642:
13563:
13251:
13132:
13097:
12795:
12669:
12462:
12361:
12178:
12086:
12007:
11851:
11745:
11694:
11612:
11547:
11477:
11365:
11283:
11195:
11125:
10954:
10884:
10753:
10661:
10516:
10330:
10235:
10162:
9941:
9768:
9653:
9556:
9515:
9326:
9175:
8933:
8883:
8794:
8678:
8634:
8571:
8453:
8383:
8288:
8192:
8090:
7960:
7927:
7845:
7765:
7585:
7482:
7280:
7225:
7079:
6966:
6892:
6859:
6804:
6649:
6553:
6492:
6357:
6267:
6212:
6112:
5973:
5822:
5698:
5599:
5538:
5534:is the Laplacian on
5436:
5339:
5306:
5163:
5095:
5041:
4961:
4901:
4812:
4764:
4673:
4610:
4538:
4469:
4351:
4287:
4187:
4085:
3955:
3922:
3827:
3803:Paley-Wiener theorem
3647:
3564:
3522:
3466:
3321:
3160:
3147:, and evaluating at
3081:
3009:
2912:
2841:
2763:
2669:
2551:
2480:
2426:
2312:
2245:
2140:
2083:
1992:
1933:
1856:
1749:
1644:
1572:
1478:
1450:by the character of
1355:the quotient of the
1290:
1137:
975:
921:
792:
769:(i.e. the points of
749:
707:. The Hilbert space
701:and μ is called the
662:
577:
534:
491:
459:
424:
413:, this *-algebra is
203:semisimple Lie group
79:; it also gives the
27238:Functional calculus
27197:Mahler's conjecture
27176:Von Neumann algebra
26890:Functional analysis
26763:Rosenberg, Jonathan
26339:1951PNAS...37..813H
25966:Kostant polynomials
25355:Harish-Chandra 1966
25281:Harish-Chandra 1952
25270:Harish-Chandra 1951
24800:
24533:
24450:
23617:. In this case the
23129:
23090:
21128:, when viewed as a
20796:
20203:
19669:
19506:
19296:reduced root system
19283:
18153:
18005:
17786:
17125:
17062:
16821:
16741:
16047:
15672:
15552:
14056:Other special cases
13948:
13457:
13354:
13305:
13180:
13122:is inverted by the
13000:
12903:
12885:
12815:
12724:
12591:
12503:
12396:
12237:
12222:
11886:
10855:is constant in the
10591:
10533:
10512:On the other hand,
10450:
10385:
10148:) is a function on
10092:
9997:
9834:
9816:
9798:
9701:
9683:
9602:
9548:Laplacian operators
8790:with determinant 1
7865:
7358:
7114:
7073:immediately yields
6988:
6716:
6353:On the other hand,
6242:
6192:
6129:
6076:
5925:
5772:
5715:
5302:) is a function on
5199:
5121:
5018:
5000:
3032:
2587:
2506:
2387:
2369:
2351:
1133:, and then setting
1080:λ is an element of
778:spherical functions
699:spherical transform
689:The transformation
446:Gelfand isomorphism
319:Spherical functions
213:by the action of a
183:spherical functions
167:corresponding to a
27263:Riemann hypothesis
26962:Topological vector
26852:10.24033/bsmf.1598
26577:, Academic Press,
26533:Helgason, Sigurdur
26516:10.1007/BF02392813
26285:Séminaire Bourbaki
26217:Gindikin, Simon G.
26142:, Academic Press,
26076:10.1007/BF01457275
25980:, pp. 452–453
25637:
25554:, pp. 423–433
25388:
25221:
25195:Schwartz functions
25179:
25123:
25027:
24924:Schwartz functions
24910:
24876:
24782:
24751:
24662:
24596:
24515:
24485:about the origin.
24451:
24432:
24415:
24372:
24283:
24102:
24054:
23964:
23578:
23514:
23421:
23283:
23197:
23130:
23111:
23091:
23072:
23026:
22936:
22816:
22716:
22603:corresponding to (
22565:
22412:
22321:
22284:
22245:
22210:
22182:
22010:
21880:
21846:
21791:
21778:
21717:
21682:
21590:
21545:
21513:
21450:
21407:fundamental domain
21395:
21355:
21332:
21268:
21232:
21146:
21114:
21004:
20990:
20930:
20884:
20821:
20797:
20782:
20744:
20690:
20659:
20624:
20566:
20470:
20444:, invariant under
20434:
20390:
20334:
20305:
20249:
20185:
20080:
20063:
19993:
19935:
19671:
19655:
19609:(λ) is defined by
19592:
19488:
19445:
19330:
19309:as above for λ in
19284:
19265:
19248:
19174:
19150:
19119:
19095:
19059:
19027:
18973:
18928:
18813:
18782:
18758:
18732:
18653:
18608:
18567:
18543:
18508:
18464:
18434:
18383:
18276:
18135:
17987:
17925:
17768:
17698:
17614:
17494:
17441:
17380:
17254:
17170:
17111:
17086:
17048:
17025:
16922:
16845:
16807:
16786:
16727:
16631:
16561:
16511:
16480:
16322:
16239:
16186:
16048:
16029:
15966:
15907:
15848:
15797:
15705:
15673:
15654:
15635:
15534:
15370:
15243:
15146:
15095:
15056:
14886:
14858:
14799:
14721:
14655:
14624:
14581:
14437:
14396:
14341:
14271:
14232:
14182:
14158:
14110:
14043:
13934:
13862:
13764:
13659:
13626:
13533:
13440:
13337:
13291:
13233:
13166:
13112:
13077:
12986:
12886:
12868:
12798:
12777:
12707:
12639:
12577:
12489:
12444:
12379:
12339:
12223:
12205:
12162:
12072:
11982:
11869:
11815:
11729:
11680:
11598:
11529:
11461:
11351:
11265:
11179:
11121:can be defined by
11070:
10936:
10811:
10724:
10629:
10577:
10519:
10500:
10436:
10371:
10296:
10217:
10124:
10075:
9980:
9903:
9820:
9802:
9784:
9750:
9687:
9669:
9635:
9588:
9533:
9471:
9233:
9156:
9147:
8917:
8869:
8863:
8766:
8757:
8651:
8604:
8555:
8437:
8369:
8268:
8174:
8072:
7944:
7894:
7851:
7825:
7731:
7620:
7569:Schwartz functions
7556:
7536:
7476:of the Laplacian:
7456:
7341:
7242:
7209:
7097:
7038:
6974:
6948:
6876:
6843:
6786:
6699:
6643:. It follows that
6611:
6525:
6476:
6341:
6321:
6253:
6228:
6196:
6178:
6171:
6115:
6108:. It follows that
6090:
6062:
5942:
5911:
5804:
5755:
5701:
5673:
5663:
5555:
5516:
5503:
5418:
5323:
5240:
5182:
5147:
5107:
5068:can be written as
5058:
5025:
5004:
4986:
4954:Laplacian operator
4940:
4883:
4781:
4740:
4655:
4649:
4604:. The real matrix
4588:
4502:
4453:
4320:
4263:
4169:
4067:
3939:
3890:
3789:
3682:
3631:Schwartz functions
3618:
3558:of the Laplacian:
3537:
3508:
3450:
3445:
3299:
3153:Plancherel formula
3137:
3046:
3018:
2993:
2894:
2823:
2743:Plancherel theorem
2729:
2627:
2570:
2535:
2492:
2453:can be written as
2443:
2410:
2373:
2355:
2337:
2305:Laplacian operator
2291:
2227:
2117:
2059:
1978:
1972:
1915:
1804:
1721:
1630:
1550:
1345:
1212:
1052:
931:
880:
788:, via the formula
759:
704:Plancherel measure
672:
587:
551:
501:
469:
434:
219:Plancherel measure
72:on the associated
70:Laplacian operator
62:Plancherel theorem
42:Plancherel formula
27340:
27339:
27243:Integral operator
27020:
27019:
26659:Knapp, Anthony W.
26297:Sternberg, Shlomo
26293:Guillemin, Victor
26003:, p. 588–589
25687:, p. 490–491
25649:orthonormal basis
25018:
24812:
24545:
24186:
24053:
23959:
23892:
23877:
23804:
23789:
23588:where α lies in Σ
23535:to the case when
22400:
22230:
22170:
22126:
21763:
21498:
21463:It follows that φ
21317:
21217:
21124:and therefore of
20975:
20869:
20609:
20523:
20331:
20135:
20124:
20041:
20039:
20003:and the constant
19933:
19866:
19851:
19778:
19763:
19644:
19518:
19376:
18429:
18368:
18165:
18017:
17798:
17730:implies that for
17680:
16720:be a function in
16298:
16249:
16230:
16177:
16161:
16096:
15919:Fourier transform
15895:
15884:
15782:
15564:
15458:
15426:
15304:
15238:
14976:
14794:
14706:
14254:with Lie algebra
14075:normal real forms
14027:
14002:
13958:
13932:
13825:
13795:
13742:
13517:
13492:
13467:
13438:
13406:
13390:
13315:
13289:
13190:
13164:
13109:
13036:
12954:
12851:
12760:
12705:
12545:
12474:
12427:
12351:so that defining
12292:
12190:
12098:
11966:
11941:
11915:
11797:
11473:As a consequence
11279:SO(2)-invariant,
11105:is a function of
10316:as a function of
8424:
8399:
7907:As a function on
7891:
7611:
7535:
7440:
7415:
7400:
7395:
7369:
7329:
7193:
7168:
7143:
7023:
6904:
6697:
6471:
6450:
6430:
6320:
6170:
5662:
5572:) commutes with Δ
5502:
4805:Riemannian metric
4799:. It carries the
4760:= SO(2), so that
4520:Example: SL(2, R)
3902:As a function on
3673:
3534:
3440:
3276:
3254:
3223:
3198:
3135:
2942:
2853:
2818:
2721:
2133:Riemannian metric
2079:= SU(2), so that
1852:upper half space
1836:Example: SL(2, C)
1124:solvable subgroup
1114:λ' is defined on
1069:In this formula:
1065:
1064:
775:positive definite
274:method of descent
187:special functions
180:positive definite
102:. In the case of
50:harmonic analysis
27370:
27330:
27329:
27248:Jones polynomial
27166:Operator algebra
26910:
26909:
26883:
26876:
26869:
26860:
26859:
26855:
26854:
26833:
26832:
26809:
26793:
26784:
26758:
26749:
26728:
26707:
26706:
26688:
26680:Kostant, Bertram
26675:
26654:
26636:Jorgenson, Jay;
26632:
26605:
26587:
26576:
26564:
26555:
26554:
26527:
26518:
26503:Acta Mathematica
26493:
26463:
26425:
26416:
26398:
26377:
26368:
26350:
26313:
26288:
26272:
26244:
26212:
26190:
26172:
26152:
26131:
26110:
26086:
26058:
26057:
26028:
26022:
26016:
26010:
26004:
25998:
25992:
25987:
25981:
25975:
25969:
25962:
25956:
25950:
25944:
25938:
25932:
25926:
25920:
25914:
25908:
25902:
25896:
25890:
25884:
25878:
25872:
25866:
25860:
25854:
25848:
25842:
25836:
25830:
25824:
25818:
25812:
25806:
25800:
25794:
25788:
25782:
25776:
25771:
25765:
25694:
25688:
25682:
25676:
25670:
25664:
25658:
25652:
25646:
25644:
25643:
25638:
25636:
25635:
25631:
25606:
25605:
25584:
25578:
25573:
25567:
25561:
25555:
25549:
25543:
25540:
25534:
25529:
25523:
25518:
25512:
25506:
25500:
25494:
25488:
25483:
25477:
25472:
25466:
25461:
25450:
25444:
25438:
25433:
25427:
25422:
25416:
25405:
25399:
25397:
25395:
25394:
25389:
25387:
25386:
25364:
25358:
25352:
25346:
25341:
25335:
25330:
25321:
25316:
25305:
25300:
25294:
25289:
25283:
25278:
25272:
25267:
25261:
25255:
25230:
25228:
25227:
25222:
25217:
25216:
25211:
25210:
25188:
25186:
25185:
25180:
25175:
25174:
25165:
25164:
25159:
25158:
25148:
25147:
25132:
25130:
25129:
25124:
25119:
25115:
25108:
25104:
25091:
25090:
25069:
25068:
25026:
25014:
25013:
25001:
24990:
24989:
24960:
24943:
24942:
24919:
24917:
24916:
24911:
24909:
24908:
24885:
24883:
24882:
24877:
24865:
24864:
24856:
24838:
24833:
24832:
24814:
24813:
24805:
24802:
24801:
24799:
24794:
24789:
24788:
24761:it follows that
24760:
24758:
24757:
24752:
24731:
24730:
24712:
24711:
24690:
24689:
24671:
24669:
24668:
24663:
24605:
24603:
24602:
24597:
24588:
24587:
24579:
24561:
24547:
24546:
24538:
24535:
24534:
24532:
24527:
24522:
24521:
24473:(λ) in terms of
24460:
24458:
24457:
24452:
24449:
24444:
24439:
24438:
24424:
24422:
24421:
24416:
24402:
24401:
24396:
24395:
24381:
24379:
24378:
24373:
24371:
24370:
24365:
24364:
24349:complexification
24334:Rosenberg (1977)
24292:
24290:
24289:
24284:
24279:
24278:
24277:
24273:
24250:
24249:
24237:
24229:
24215:
24214:
24202:
24188:
24187:
24179:
24176:
24111:
24109:
24108:
24103:
24098:
24094:
24084:
24083:
24068:
24067:
24055:
24046:
24034:
24033:
24032:
24031:
24013:
24008:
24007:
23990:
23989:
23973:
23971:
23970:
23965:
23960:
23958:
23957:
23953:
23952:
23948:
23944:
23943:
23919:
23918:
23903:
23902:
23893:
23885:
23878:
23870:
23859:
23855:
23854:
23850:
23846:
23845:
23815:
23814:
23805:
23797:
23790:
23782:
23770:
23763:
23762:
23735:
23734:
23730:
23729:
23699:
23697:
23696:
23675:
23674:
23673:
23672:
23625:by noting that φ
23587:
23585:
23584:
23579:
23577:
23576:
23568:
23567:
23523:
23521:
23520:
23515:
23507:
23506:
23485:
23484:
23463:
23462:
23453:
23452:
23430:
23428:
23427:
23422:
23411:
23410:
23409:
23408:
23388:
23387:
23375:
23374:
23373:
23372:
23346:
23345:
23344:
23343:
23334:
23333:
23292:
23290:
23289:
23284:
23279:
23278:
23260:
23259:
23247:
23246:
23206:
23204:
23203:
23198:
23184:
23183:
23182:
23181:
23139:
23137:
23136:
23131:
23128:
23123:
23118:
23117:
23100:
23098:
23097:
23092:
23089:
23084:
23079:
23078:
23035:
23033:
23032:
23027:
23022:
23021:
23000:
22999:
22978:
22977:
22968:
22967:
22945:
22943:
22942:
22937:
22923:
22922:
22901:
22900:
22888:
22887:
22860:
22859:
22850:
22849:
22825:
22823:
22822:
22817:
22785:
22784:
22760:
22759:
22725:
22723:
22722:
22717:
22687:
22686:
22679:
22678:
22574:
22572:
22571:
22566:
22564:
22563:
22558:
22557:
22443:are surveyed in
22421:
22419:
22418:
22413:
22401:
22396:
22386:
22377:
22375:
22374:
22330:
22328:
22327:
22322:
22320:
22319:
22314:
22313:
22293:
22291:
22290:
22285:
22277:
22276:
22258:
22257:
22244:
22219:
22217:
22216:
22211:
22209:
22208:
22191:
22189:
22188:
22183:
22171:
22169:
22165:
22164:
22149:
22148:
22136:
22127:
22122:
22112:
22103:
22101:
22099:
22098:
22080:
22079:
22055:
22054:
22042:
22041:
22019:
22017:
22016:
22011:
21999:
21998:
21977:
21969:
21968:
21964:
21939:
21938:
21889:
21887:
21886:
21881:
21879:
21878:
21873:
21872:
21855:
21853:
21852:
21847:
21845:
21844:
21839:
21838:
21800:
21798:
21797:
21792:
21777:
21726:
21724:
21723:
21718:
21716:
21715:
21710:
21709:
21691:
21689:
21688:
21683:
21681:
21680:
21629:
21628:
21616:
21615:
21599:
21597:
21596:
21591:
21589:
21588:
21583:
21582:
21554:
21552:
21551:
21546:
21541:
21540:
21512:
21494:
21493:
21459:
21457:
21456:
21451:
21449:
21448:
21404:
21402:
21401:
21396:
21394:
21393:
21388:
21387:
21364:
21362:
21361:
21356:
21351:
21350:
21331:
21289:. The expansion
21277:
21275:
21274:
21269:
21264:
21263:
21242:
21241:
21231:
21216:
21215:
21194:
21193:
21155:
21153:
21152:
21147:
21145:
21144:
21123:
21121:
21120:
21115:
21113:
21112:
21103:
21099:
21098:
21097:
21092:
21077:
21076:
21071:
21051:
21050:
21013:
21011:
21010:
21005:
21000:
20999:
20989:
20971:
20970:
20969:
20956:
20955:
20939:
20937:
20936:
20931:
20926:
20925:
20894:
20893:
20883:
20865:
20864:
20830:
20828:
20827:
20822:
20820:
20819:
20806:
20804:
20803:
20798:
20795:
20790:
20772:
20771:
20770:
20753:
20751:
20750:
20745:
20719:
20718:
20699:
20697:
20696:
20691:
20689:
20688:
20668:
20666:
20665:
20660:
20655:
20654:
20634:
20633:
20623:
20605:
20604:
20603:
20575:
20573:
20572:
20567:
20562:
20561:
20550:
20546:
20524:
20522:
20511:
20479:
20477:
20476:
20471:
20469:
20468:
20452:. In general if
20443:
20441:
20440:
20435:
20433:
20432:
20399:
20397:
20396:
20391:
20389:
20388:
20343:
20341:
20340:
20335:
20333:
20332:
20324:
20314:
20312:
20311:
20306:
20298:
20281:
20280:
20258:
20256:
20255:
20250:
20238:
20237:
20229:
20211:
20202:
20197:
20192:
20191:
20181:
20180:
20162:
20145:
20144:
20133:
20126:
20125:
20117:
20089:
20087:
20086:
20081:
20073:
20072:
20062:
20061:
20060:
20040:
20032:
20002:
20000:
19999:
19994:
19989:
19988:
19961:
19960:
19944:
19942:
19941:
19936:
19934:
19932:
19931:
19927:
19926:
19922:
19918:
19917:
19893:
19892:
19877:
19876:
19867:
19859:
19852:
19844:
19833:
19829:
19828:
19824:
19820:
19819:
19789:
19788:
19779:
19771:
19764:
19756:
19744:
19737:
19736:
19709:
19708:
19704:
19703:
19673:
19670:
19668:
19663:
19640:
19639:
19601:
19599:
19598:
19593:
19581:
19580:
19572:
19554:
19539:
19538:
19520:
19519:
19511:
19508:
19507:
19505:
19500:
19495:
19494:
19454:
19452:
19451:
19446:
19425:
19424:
19400:
19399:
19378:
19377:
19369:
19360:) is defined by
19339:
19337:
19336:
19331:
19329:
19328:
19323:
19322:
19293:
19291:
19290:
19285:
19282:
19277:
19272:
19271:
19257:
19255:
19254:
19249:
19238:
19237:
19228:
19214:
19213:
19183:
19181:
19180:
19175:
19173:
19172:
19159:
19157:
19156:
19151:
19149:
19148:
19143:
19142:
19128:
19126:
19125:
19120:
19118:
19117:
19104:
19102:
19101:
19096:
19094:
19093:
19068:
19066:
19065:
19060:
19058:
19057:
19036:
19034:
19033:
19028:
19026:
19025:
19020:
19019:
19003:
19002:
18982:
18980:
18979:
18974:
18960:
18959:
18954:
18953:
18937:
18935:
18934:
18929:
18918:
18917:
18864:
18863:
18845:
18844:
18839:
18838:
18822:
18820:
18819:
18814:
18812:
18811:
18806:
18805:
18791:
18789:
18788:
18783:
18781:
18780:
18767:
18765:
18764:
18759:
18757:
18756:
18741:
18739:
18738:
18733:
18722:
18721:
18698:
18697:
18688:
18687:
18678:
18677:
18662:
18660:
18659:
18654:
18652:
18651:
18646:
18645:
18635:
18634:
18617:
18615:
18614:
18609:
18607:
18606:
18601:
18600:
18590:
18589:
18576:
18574:
18573:
18568:
18566:
18565:
18552:
18550:
18549:
18544:
18542:
18541:
18536:
18535:
18517:
18515:
18514:
18509:
18507:
18506:
18473:
18471:
18470:
18465:
18463:
18462:
18443:
18441:
18440:
18435:
18430:
18428:
18411:
18391:
18382:
18364:
18363:
18285:
18283:
18282:
18277:
18265:
18264:
18259:
18238:
18233:
18232:
18227:
18226:
18186:
18185:
18167:
18166:
18158:
18155:
18154:
18152:
18147:
18142:
18141:
18119:
18118:
18113:
18092:
18087:
18086:
18081:
18080:
18054:
18053:
18041:
18040:
18019:
18018:
18010:
18007:
18006:
18004:
17999:
17994:
17993:
17966:
17965:
17934:
17932:
17931:
17926:
17914:
17913:
17908:
17887:
17879:
17862:
17854:
17853:
17848:
17847:
17819:
17818:
17800:
17799:
17791:
17788:
17787:
17785:
17780:
17775:
17774:
17707:
17705:
17704:
17699:
17682:
17681:
17673:
17658:
17657:
17645:
17644:
17623:
17621:
17620:
17615:
17594:
17593:
17581:
17580:
17559:
17558:
17546:
17545:
17503:
17501:
17500:
17495:
17493:
17492:
17480:
17479:
17450:
17448:
17447:
17442:
17437:
17436:
17418:
17417:
17408:
17407:
17389:
17387:
17386:
17381:
17370:
17369:
17351:
17350:
17349:
17348:
17339:
17334:
17333:
17298:
17297:
17293:
17263:
17261:
17260:
17255:
17228:
17227:
17212:
17211:
17196:
17195:
17179:
17177:
17176:
17171:
17166:
17165:
17156:
17151:
17150:
17138:
17137:
17124:
17119:
17095:
17093:
17092:
17087:
17079:
17061:
17056:
17034:
17032:
17031:
17026:
17018:
17017:
16931:
16929:
16928:
16923:
16908:
16907:
16889:
16888:
16854:
16852:
16851:
16846:
16838:
16820:
16815:
16795:
16793:
16792:
16787:
16782:
16781:
16772:
16767:
16766:
16754:
16753:
16740:
16735:
16640:
16638:
16637:
16632:
16630:
16629:
16617:
16616:
16607:
16602:
16601:
16589:
16588:
16570:
16568:
16567:
16562:
16557:
16556:
16541:
16520:
16518:
16517:
16512:
16510:
16509:
16489:
16487:
16486:
16481:
16473:
16472:
16351:normal real form
16331:
16329:
16328:
16323:
16318:
16317:
16299:
16297:
16283:
16273:
16272:
16262:
16251:
16250:
16242:
16238:
16195:
16193:
16192:
16187:
16178:
16176:
16162:
16157:
16147:
16146:
16136:
16134:
16120:
16119:
16098:
16097:
16089:
16058:and the role of
16057:
16055:
16054:
16049:
16046:
16041:
16036:
16035:
15975:
15973:
15972:
15967:
15962:
15961:
15943:
15942:
15916:
15914:
15913:
15908:
15897:
15896:
15888:
15885:
15880:
15866:
15857:
15855:
15854:
15849:
15844:
15843:
15813:
15812:
15796:
15775:
15774:
15762:
15761:
15749:
15748:
15714:
15712:
15711:
15706:
15704:
15703:
15682:
15680:
15679:
15674:
15671:
15666:
15661:
15660:
15644:
15642:
15641:
15636:
15623:
15622:
15617:
15599:
15585:
15584:
15566:
15565:
15557:
15554:
15553:
15551:
15546:
15541:
15540:
15517:
15516:
15511:
15493:
15479:
15478:
15460:
15459:
15451:
15448:
15447:
15446:
15445:
15440:
15439:
15427:
15425:
15424:
15416:
15407:
15379:
15377:
15376:
15371:
15353:
15352:
15328:
15327:
15306:
15305:
15297:
15252:
15250:
15249:
15244:
15239:
15237:
15223:
15219:
15218:
15206:
15205:
15195:
15187:
15186:
15174:
15173:
15155:
15153:
15152:
15147:
15145:
15144:
15139:
15138:
15125:
15124:
15104:
15102:
15101:
15096:
15094:
15093:
15081:
15080:
15065:
15063:
15062:
15057:
15045:
15044:
15039:
15030:
15029:
15014:
15005:
15004:
14989:
14988:
14987:
14977:
14975:
14974:
14966:
14957:
14933:
14932:
14895:
14893:
14892:
14887:
14885:
14884:
14867:
14865:
14864:
14859:
14854:
14840:
14839:
14808:
14806:
14805:
14800:
14795:
14793:
14789:
14788:
14772:
14771:
14770:
14737:
14736:
14720:
14704:
14696:
14695:
14683:
14682:
14664:
14662:
14661:
14656:
14654:
14653:
14648:
14647:
14633:
14631:
14630:
14625:
14623:
14622:
14613:
14612:
14607:
14606:
14590:
14588:
14587:
14582:
14577:
14576:
14539:
14538:
14517:
14516:
14504:
14503:
14482:
14481:
14469:
14468:
14446:
14444:
14443:
14438:
14436:
14435:
14430:
14429:
14405:
14403:
14402:
14397:
14350:
14348:
14347:
14342:
14337:
14336:
14311:
14310:
14280:
14278:
14277:
14272:
14267:
14266:
14241:
14239:
14238:
14233:
14228:
14227:
14215:
14214:
14205:
14204:
14191:
14189:
14188:
14183:
14181:
14180:
14167:
14165:
14164:
14159:
14157:
14156:
14138:complexification
14119:
14117:
14116:
14111:
14109:
14108:
14092:The approach of
14052:
14050:
14049:
14044:
14032:
14028:
14023:
14015:
14003:
13998:
13990:
13979:
13978:
13960:
13959:
13951:
13947:
13942:
13933:
13931:
13930:
13918:
13871:
13869:
13868:
13863:
13849:
13848:
13827:
13826:
13818:
13803:
13802:
13797:
13796:
13788:
13773:
13771:
13770:
13765:
13763:
13762:
13744:
13743:
13735:
13729:
13728:
13710:
13709:
13668:
13666:
13665:
13660:
13658:
13657:
13652:
13651:
13635:
13633:
13632:
13627:
13597:
13596:
13575:
13574:
13549:. Now for fixed
13542:
13540:
13539:
13534:
13522:
13518:
13513:
13505:
13493:
13488:
13480:
13469:
13468:
13460:
13456:
13451:
13439:
13437:
13436:
13435:
13419:
13407:
13399:
13391:
13389:
13378:
13374:
13356:
13353:
13348:
13317:
13316:
13308:
13304:
13299:
13290:
13288:
13287:
13286:
13270:
13242:
13240:
13239:
13234:
13192:
13191:
13183:
13179:
13174:
13165:
13157:
13121:
13119:
13118:
13113:
13111:
13110:
13102:
13086:
13084:
13083:
13078:
13042:
13038:
13037:
13032:
13031:
13022:
13008:
12999:
12994:
12985:
12960:
12956:
12955:
12950:
12949:
12948:
12936:
12935:
12925:
12911:
12902:
12897:
12884:
12879:
12857:
12853:
12852:
12847:
12846:
12837:
12823:
12814:
12809:
12786:
12784:
12783:
12778:
12766:
12762:
12761:
12756:
12755:
12746:
12732:
12723:
12718:
12706:
12704:
12696:
12688:
12648:
12646:
12645:
12640:
12628:
12627:
12590:
12585:
12566:
12565:
12550:
12546:
12541:
12540:
12539:
12524:
12523:
12513:
12502:
12497:
12476:
12475:
12467:
12453:
12451:
12450:
12445:
12433:
12429:
12428:
12423:
12422:
12413:
12395:
12390:
12348:
12346:
12345:
12340:
12321:
12320:
12316:
12297:
12293:
12288:
12287:
12286:
12274:
12273:
12258:
12257:
12247:
12236:
12231:
12221:
12216:
12192:
12191:
12183:
12171:
12169:
12168:
12163:
12142:
12122:
12121:
12100:
12099:
12091:
12081:
12079:
12078:
12073:
12049:
12041:
12040:
12019:
12018:
11991:
11989:
11988:
11983:
11971:
11967:
11962:
11954:
11942:
11937:
11929:
11917:
11916:
11908:
11896:
11895:
11885:
11880:
11824:
11822:
11821:
11816:
11799:
11798:
11790:
11775:
11774:
11769:
11762:
11761:
11741:It follows that
11738:
11736:
11735:
11730:
11725:
11724:
11709:
11708:
11689:
11687:
11686:
11681:
11652:
11651:
11639:
11638:
11607:
11605:
11604:
11599:
11597:
11596:
11572:
11571:
11559:
11558:
11538:
11536:
11535:
11530:
11525:
11524:
11515:
11514:
11499:
11498:
11489:
11488:
11470:
11468:
11467:
11462:
11429:
11428:
11427:
11426:
11384:
11383:
11382:
11381:
11360:
11358:
11357:
11352:
11341:
11340:
11295:
11294:
11274:
11272:
11271:
11266:
11254:
11253:
11244:
11243:
11242:
11241:
11214:
11213:
11212:
11211:
11188:
11186:
11185:
11180:
11169:
11137:
11136:
11079:
11077:
11076:
11071:
11053:
11052:
11048:
11035:
11034:
11022:
11021:
11003:
11002:
11001:
11000:
10973:
10972:
10971:
10970:
10945:
10943:
10942:
10937:
10928:
10927:
10912:
10911:
10910:
10909:
10820:
10818:
10817:
10812:
10804:
10803:
10794:
10793:
10775:
10774:
10765:
10764:
10736:descent relation
10733:
10731:
10730:
10725:
10717:
10716:
10689:
10688:
10676:
10675:
10638:
10636:
10635:
10630:
10625:
10624:
10590:
10585:
10573:
10572:
10532:
10527:
10509:
10507:
10506:
10501:
10493:
10492:
10449:
10444:
10429:
10428:
10410:
10409:
10384:
10379:
10361:
10360:
10348:
10347:
10305:
10303:
10302:
10297:
10285:
10284:
10272:
10271:
10247:
10246:
10226:
10224:
10223:
10218:
10133:
10131:
10130:
10125:
10091:
10086:
10055:
10054:
10053:
10052:
10019:
10018:
9996:
9991:
9960:
9959:
9958:
9957:
9912:
9910:
9909:
9904:
9898:
9897:
9882:
9881:
9866:
9865:
9850:
9849:
9833:
9828:
9815:
9810:
9797:
9792:
9780:
9779:
9759:
9757:
9756:
9751:
9749:
9748:
9733:
9732:
9717:
9716:
9700:
9695:
9682:
9677:
9665:
9664:
9644:
9642:
9641:
9636:
9630:
9629:
9601:
9596:
9584:
9583:
9568:
9567:
9542:
9540:
9539:
9534:
9480:
9478:
9477:
9472:
9453:
9452:
9448:
9432:
9431:
9377:
9376:
9372:
9356:
9355:
9242:
9240:
9239:
9234:
9232:
9231:
9219:
9218:
9206:
9205:
9187:
9186:
9165:
9163:
9162:
9157:
9152:
9151:
8945:
8944:
8926:
8924:
8923:
8918:
8913:
8912:
8878:
8876:
8875:
8870:
8868:
8867:
8775:
8773:
8772:
8767:
8762:
8761:
8690:
8689:
8660:
8658:
8657:
8652:
8650:
8649:
8644:
8643:
8613:
8611:
8610:
8605:
8600:
8599:
8594:
8593:
8583:
8582:
8564:
8562:
8561:
8556:
8538:
8537:
8519:
8518:
8497:
8496:
8492:
8446:
8444:
8443:
8438:
8429:
8425:
8420:
8412:
8400:
8395:
8387:
8378:
8376:
8375:
8370:
8365:
8339:
8338:
8323:
8322:
8317:
8316:
8306:
8305:
8277:
8275:
8274:
8269:
8261:
8260:
8248:
8247:
8229:
8228:
8204:
8203:
8183:
8181:
8180:
8175:
8173:
8172:
8164:
8160:
8159:
8158:
8153:
8144:
8127:
8126:
8102:
8101:
8081:
8079:
8078:
8073:
8047:
8046:
8029:
8011:
7988:
7987:
7972:
7971:
7953:
7951:
7950:
7945:
7943:
7942:
7937:
7936:
7903:
7901:
7900:
7895:
7893:
7892:
7884:
7878:
7877:
7864:
7859:
7834:
7832:
7831:
7826:
7821:
7820:
7819:
7808:
7789:
7772:
7740:
7738:
7737:
7732:
7727:
7724:
7723:
7719:
7712:
7708:
7698:
7697:
7676:
7675:
7654:
7653:
7644:
7643:
7619:
7594:
7593:
7565:
7563:
7562:
7557:
7552:
7551:
7542:
7538:
7537:
7528:
7522:
7521:
7504:
7503:
7494:
7493:
7465:
7463:
7462:
7457:
7445:
7441:
7436:
7428:
7416:
7411:
7403:
7401:
7396:
7391:
7390:
7381:
7379:
7377:
7376:
7371:
7370:
7362:
7357:
7352:
7330:
7325:
7324:
7315:
7313:
7312:
7303:
7302:
7301:
7300:
7295:
7294:
7251:
7249:
7248:
7243:
7241:
7240:
7235:
7234:
7218:
7216:
7215:
7210:
7198:
7194:
7189:
7181:
7169:
7164:
7156:
7145:
7144:
7136:
7124:
7123:
7113:
7108:
7047:
7045:
7044:
7039:
7025:
7024:
7016:
7001:
7000:
6995:
6987:
6982:
6957:
6955:
6954:
6949:
6937:
6936:
6906:
6905:
6897:
6885:
6883:
6882:
6877:
6875:
6874:
6869:
6868:
6852:
6850:
6849:
6844:
6839:
6838:
6829:
6828:
6816:
6815:
6795:
6793:
6792:
6787:
6775:
6774:
6757:
6753:
6715:
6707:
6698:
6696:
6685:
6674:
6673:
6661:
6660:
6620:
6618:
6617:
6612:
6610:
6609:
6600:
6599:
6590:
6589:
6565:
6564:
6534:
6532:
6531:
6526:
6515:
6510:
6509:
6485:
6483:
6482:
6477:
6472:
6467:
6459:
6451:
6443:
6431:
6429:
6415:
6401:
6384:
6383:
6350:
6348:
6347:
6342:
6337:
6336:
6327:
6323:
6322:
6313:
6307:
6306:
6289:
6288:
6279:
6278:
6262:
6260:
6259:
6254:
6252:
6251:
6241:
6236:
6224:
6223:
6205:
6203:
6202:
6197:
6191:
6186:
6177:
6173:
6172:
6163:
6157:
6156:
6139:
6138:
6128:
6123:
6099:
6097:
6096:
6091:
6075:
6070:
6052:
6051:
6050:
6049:
6044:
6043:
6009:
6008:
5996:
5995:
5994:
5993:
5988:
5987:
5951:
5949:
5948:
5943:
5924:
5919:
5901:
5900:
5899:
5898:
5893:
5892:
5858:
5857:
5845:
5844:
5843:
5842:
5837:
5836:
5813:
5811:
5810:
5805:
5771:
5766:
5754:
5753:
5749:
5714:
5709:
5682:
5680:
5679:
5674:
5669:
5665:
5664:
5655:
5649:
5648:
5634:
5633:
5621:
5620:
5611:
5610:
5564:
5562:
5561:
5556:
5554:
5553:
5548:
5547:
5525:
5523:
5522:
5517:
5509:
5505:
5504:
5495:
5489:
5488:
5474:
5473:
5458:
5457:
5448:
5447:
5427:
5425:
5424:
5419:
5396:
5395:
5391:
5351:
5350:
5332:
5330:
5329:
5324:
5322:
5321:
5316:
5315:
5249:
5247:
5246:
5241:
5230:
5226:
5198:
5193:
5156:
5154:
5153:
5148:
5143:
5142:
5120:
5115:
5067:
5065:
5064:
5059:
5057:
5056:
5051:
5050:
5034:
5032:
5031:
5026:
5017:
5012:
4999:
4994:
4982:
4981:
4949:
4947:
4946:
4941:
4925:
4924:
4892:
4890:
4889:
4884:
4882:
4878:
4877:
4876:
4861:
4860:
4843:
4842:
4827:
4826:
4790:
4788:
4787:
4782:
4780:
4779:
4774:
4773:
4749:
4747:
4746:
4741:
4736:
4735:
4664:
4662:
4661:
4656:
4654:
4653:
4597:
4595:
4594:
4589:
4554:
4553:
4548:
4547:
4511:
4509:
4508:
4503:
4498:
4497:
4492:
4491:
4481:
4480:
4462:
4460:
4459:
4454:
4436:
4435:
4417:
4416:
4395:
4394:
4390:
4341:(with measure λ
4340:
4329:
4327:
4326:
4321:
4316:
4315:
4310:
4309:
4299:
4298:
4272:
4270:
4269:
4264:
4256:
4255:
4243:
4242:
4224:
4223:
4199:
4198:
4178:
4176:
4175:
4170:
4168:
4167:
4159:
4155:
4154:
4153:
4148:
4139:
4122:
4121:
4097:
4096:
4076:
4074:
4073:
4068:
4042:
4041:
4024:
4006:
3983:
3982:
3967:
3966:
3948:
3946:
3945:
3940:
3938:
3937:
3932:
3931:
3899:
3897:
3896:
3891:
3886:
3885:
3884:
3880:
3851:
3834:
3798:
3796:
3795:
3790:
3785:
3782:
3781:
3777:
3770:
3766:
3738:
3737:
3716:
3715:
3706:
3705:
3681:
3656:
3655:
3627:
3625:
3624:
3619:
3614:
3613:
3595:
3594:
3579:
3578:
3546:
3544:
3543:
3538:
3536:
3535:
3527:
3517:
3515:
3514:
3509:
3501:
3484:
3483:
3459:
3457:
3456:
3451:
3449:
3448:
3442:
3441:
3433:
3404:
3403:
3391:
3383:
3382:
3364:
3347:
3346:
3308:
3306:
3305:
3300:
3288:
3287:
3277:
3272:
3262:
3261:
3256:
3255:
3247:
3242:
3231:
3230:
3225:
3224:
3216:
3199:
3194:
3193:
3184:
3182:
3181:
3172:
3171:
3146:
3144:
3143:
3138:
3136:
3131:
3127:
3126:
3107:
3093:
3092:
3055:
3053:
3052:
3047:
3045:
3044:
3031:
3026:
3002:
3000:
2999:
2994:
2982:
2981:
2963:
2962:
2944:
2943:
2935:
2903:
2901:
2900:
2895:
2886:
2885:
2855:
2854:
2846:
2832:
2830:
2829:
2824:
2819:
2817:
2803:
2789:
2775:
2774:
2738:
2736:
2735:
2730:
2722:
2720:
2719:
2718:
2705:
2704:
2695:
2681:
2680:
2636:
2634:
2633:
2628:
2610:
2609:
2586:
2581:
2544:
2542:
2541:
2536:
2531:
2530:
2505:
2500:
2452:
2450:
2449:
2444:
2442:
2441:
2436:
2435:
2419:
2417:
2416:
2411:
2406:
2405:
2386:
2381:
2368:
2363:
2350:
2345:
2333:
2332:
2300:
2298:
2297:
2292:
2269:
2268:
2236:
2234:
2233:
2228:
2226:
2222:
2221:
2220:
2205:
2204:
2189:
2188:
2171:
2170:
2155:
2154:
2126:
2124:
2123:
2118:
2110:
2099:
2098:
2093:
2092:
2068:
2066:
2065:
2060:
2055:
2054:
1987:
1985:
1984:
1979:
1977:
1976:
1924:
1922:
1921:
1916:
1872:
1871:
1866:
1865:
1813:
1811:
1810:
1805:
1761:
1760:
1730:
1728:
1727:
1722:
1704:
1703:
1698:
1680:
1675:
1674:
1662:
1661:
1639:
1637:
1636:
1631:
1620:
1619:
1559:
1557:
1556:
1551:
1525:
1524:
1520:
1379:It follows that
1354:
1352:
1351:
1346:
1332:
1331:
1322:
1308:
1307:
1246:modular function
1221:
1219:
1218:
1213:
1199:
1198:
1194:
1175:
1174:
1147:
1061:
1059:
1058:
1053:
1041:
1040:
1019:
1018:
1009:
1008:
987:
986:
969:
940:
938:
937:
932:
930:
929:
889:
887:
886:
881:
860:
859:
835:
834:
804:
803:
768:
766:
765:
760:
758:
757:
741:The characters χ
730:), the space of
681:
679:
678:
673:
671:
670:
616:it generates on
612:. Moreover, the
596:
594:
593:
588:
586:
585:
560:
558:
557:
552:
550:
549:
544:
543:
510:
508:
507:
502:
500:
499:
478:
476:
475:
470:
468:
467:
443:
441:
440:
435:
433:
432:
291:. The classical
280:. In particular
278:Jacques Hadamard
266:hyperbolic space
104:hyperbolic space
101:
27378:
27377:
27373:
27372:
27371:
27369:
27368:
27367:
27343:
27342:
27341:
27336:
27318:
27282:Advanced topics
27277:
27201:
27180:
27139:
27105:Hilbert–Schmidt
27078:
27069:Gelfand–Naimark
27016:
26966:
26901:
26887:
26747:
26726:
26686:
26673:
26652:
26603:
26585:
26483:10.2307/2372772
26445:10.2307/2372786
26333:(12): 813–818,
26311:
26277:Godement, Roger
26226:
26188:
26158:J. Funct. Anal.
26150:
26136:Dieudonné, Jean
26129:
26042:J. Funct. Anal.
26036:
26031:
26023:
26019:
26011:
26007:
25999:
25995:
25988:
25984:
25976:
25972:
25963:
25959:
25951:
25947:
25939:
25935:
25927:
25923:
25915:
25911:
25903:
25899:
25891:
25887:
25879:
25875:
25867:
25863:
25855:
25851:
25843:
25839:
25831:
25827:
25819:
25815:
25807:
25803:
25795:
25791:
25783:
25779:
25772:
25768:
25762:Gindikin (2008)
25754:Beerends (1987)
25750:Radon transform
25747:
25728:
25722:
25716:
25705:
25695:
25691:
25683:
25679:
25671:
25667:
25659:
25655:
25627:
25620:
25616:
25601:
25597:
25595:
25592:
25591:
25585:
25581:
25574:
25570:
25562:
25558:
25550:
25546:
25541:
25537:
25530:
25526:
25519:
25515:
25507:
25503:
25495:
25491:
25484:
25480:
25473:
25469:
25462:
25453:
25445:
25441:
25434:
25430:
25423:
25419:
25406:
25402:
25382:
25381:
25379:
25376:
25375:
25365:
25361:
25353:
25349:
25342:
25338:
25331:
25324:
25317:
25308:
25301:
25297:
25290:
25286:
25279:
25275:
25268:
25264:
25256:
25252:
25248:
25212:
25206:
25205:
25204:
25202:
25199:
25198:
25170:
25166:
25160:
25154:
25153:
25152:
25143:
25142:
25140:
25137:
25136:
25086:
25082:
25064:
25060:
25032:
25028:
25022:
25009:
25005:
24997:
24985:
24981:
24974:
24970:
24956:
24938:
24937:
24935:
24932:
24931:
24926:
24904:
24903:
24901:
24898:
24897:
24857:
24852:
24851:
24834:
24828:
24824:
24804:
24803:
24795:
24790:
24784:
24783:
24781:
24777:
24766:
24763:
24762:
24723:
24719:
24707:
24703:
24685:
24681:
24679:
24676:
24675:
24627:
24624:
24623:
24580:
24575:
24574:
24557:
24537:
24536:
24528:
24523:
24517:
24516:
24514:
24510:
24493:
24490:
24489:
24475:Gamma functions
24468:
24445:
24440:
24434:
24433:
24430:
24427:
24426:
24397:
24391:
24390:
24389:
24387:
24384:
24383:
24366:
24360:
24359:
24358:
24356:
24353:
24352:
24331:
24316:Helgason (1970)
24263:
24259:
24255:
24251:
24242:
24238:
24233:
24225:
24210:
24206:
24198:
24178:
24177:
24172:
24170:
24167:
24166:
24137:
24130:
24076:
24072:
24063:
24059:
24044:
24043:
24039:
24024:
24020:
24009:
24003:
23999:
23998:
23994:
23985:
23981:
23979:
23976:
23975:
23939:
23935:
23911:
23907:
23898:
23894:
23884:
23883:
23879:
23869:
23868:
23864:
23841:
23837:
23810:
23806:
23796:
23795:
23791:
23781:
23780:
23776:
23771:
23758:
23754:
23725:
23721:
23705:
23701:
23700:
23698:
23692:
23688:
23668:
23664:
23663:
23659:
23657:
23654:
23653:
23628:
23608:
23591:
23569:
23563:
23562:
23561:
23559:
23556:
23555:
23545:
23534:
23502:
23498:
23480:
23476:
23458:
23454:
23448:
23444:
23436:
23433:
23432:
23404:
23400:
23399:
23395:
23383:
23379:
23368:
23364:
23363:
23359:
23339:
23335:
23329:
23325:
23324:
23320:
23318:
23315:
23314:
23296:
23274:
23270:
23255:
23251:
23242:
23238:
23218:
23215:
23214:
23177:
23173:
23172:
23168:
23151:
23148:
23147:
23124:
23119:
23113:
23112:
23106:
23103:
23102:
23085:
23080:
23074:
23073:
23070:
23067:
23066:
23064:
23017:
23013:
22995:
22991:
22973:
22969:
22963:
22959:
22951:
22948:
22947:
22918:
22914:
22896:
22892:
22883:
22879:
22855:
22851:
22845:
22841:
22833:
22830:
22829:
22777:
22773:
22755:
22751:
22731:
22728:
22727:
22671:
22667:
22651:
22647:
22612:
22609:
22608:
22559:
22553:
22552:
22551:
22549:
22546:
22545:
22511:
22498:
22486:
22478:
22461:
22454:
22445:Helgason (2000)
22433:
22427:
22379:
22378:
22376:
22361:
22357:
22340:
22337:
22336:
22334:
22315:
22309:
22308:
22307:
22305:
22302:
22301:
22266:
22262:
22250:
22246:
22234:
22228:
22225:
22224:
22204:
22203:
22201:
22198:
22197:
22157:
22153:
22144:
22140:
22129:
22128:
22105:
22104:
22102:
22100:
22085:
22081:
22066:
22062:
22050:
22046:
22037:
22033:
22031:
22028:
22027:
21991:
21987:
21970:
21960:
21956:
21952:
21934:
21930:
21928:
21925:
21924:
21913:
21874:
21868:
21867:
21866:
21861:
21858:
21857:
21840:
21834:
21833:
21832:
21830:
21827:
21826:
21820:
21813:Coxeter element
21767:
21755:
21752:
21751:
21737:
21711:
21705:
21704:
21703:
21701:
21698:
21697:
21655:
21651:
21624:
21620:
21611:
21607:
21605:
21602:
21601:
21584:
21578:
21577:
21576:
21574:
21571:
21570:
21568:
21533:
21529:
21502:
21489:
21485:
21483:
21480:
21479:
21473:
21466:
21444:
21443:
21441:
21438:
21437:
21419:
21389:
21383:
21382:
21381:
21379:
21376:
21375:
21373:
21337:
21333:
21321:
21294:
21291:
21290:
21284:
21256:
21252:
21237:
21233:
21221:
21202:
21198:
21189:
21185:
21183:
21180:
21179:
21177:
21173:
21140:
21139:
21137:
21134:
21133:
21108:
21104:
21093:
21082:
21081:
21072:
21061:
21060:
21059:
21055:
21046:
21042:
21037:
21034:
21033:
21031:
21024:
20995:
20991:
20979:
20965:
20964:
20960:
20951:
20947:
20945:
20942:
20941:
20921:
20917:
20889:
20885:
20873:
20860:
20856:
20848:
20845:
20844:
20839:
20815:
20814:
20812:
20809:
20808:
20791:
20786:
20766:
20765:
20761:
20759:
20756:
20755:
20714:
20710:
20705:
20702:
20701:
20684:
20683:
20681:
20678:
20677:
20675:
20650:
20646:
20629:
20625:
20613:
20599:
20598:
20594:
20586:
20583:
20582:
20551:
20515:
20510:
20509:
20506:
20505:
20485:
20482:
20481:
20464:
20463:
20461:
20458:
20457:
20428:
20427:
20425:
20422:
20421:
20384:
20383:
20381:
20378:
20377:
20360:, functions on
20350:
20323:
20322:
20320:
20317:
20316:
20294:
20276:
20272:
20264:
20261:
20260:
20230:
20225:
20224:
20207:
20198:
20193:
20187:
20186:
20176:
20172:
20158:
20140:
20136:
20116:
20115:
20101:
20098:
20097:
20068:
20064:
20056:
20052:
20045:
20031:
20023:
20020:
20019:
20010:chosen so that
20009:
19981:
19977:
19956:
19952:
19950:
19947:
19946:
19913:
19909:
19885:
19881:
19872:
19868:
19858:
19857:
19853:
19843:
19842:
19838:
19815:
19811:
19784:
19780:
19770:
19769:
19765:
19755:
19754:
19750:
19745:
19732:
19728:
19699:
19695:
19679:
19675:
19674:
19672:
19664:
19659:
19648:
19635:
19631:
19614:
19611:
19610:
19573:
19568:
19567:
19550:
19534:
19530:
19510:
19509:
19501:
19496:
19490:
19489:
19487:
19483:
19466:
19463:
19462:
19417:
19413:
19395:
19391:
19368:
19367:
19365:
19362:
19361:
19347:
19324:
19318:
19317:
19316:
19314:
19311:
19310:
19308:
19301:
19278:
19273:
19267:
19266:
19263:
19260:
19259:
19233:
19229:
19224:
19209:
19205:
19197:
19194:
19193:
19168:
19167:
19165:
19162:
19161:
19144:
19138:
19137:
19136:
19134:
19131:
19130:
19129:, which allows
19113:
19112:
19110:
19107:
19106:
19089:
19088:
19086:
19083:
19082:
19053:
19052:
19050:
19047:
19046:
19021:
19015:
19014:
19013:
18998:
18994:
18992:
18989:
18988:
18985:restricted root
18955:
18949:
18948:
18947:
18945:
18942:
18941:
18913:
18912:
18859:
18858:
18840:
18834:
18833:
18832:
18830:
18827:
18826:
18807:
18801:
18800:
18799:
18797:
18794:
18793:
18776:
18775:
18773:
18770:
18769:
18752:
18751:
18749:
18746:
18745:
18717:
18716:
18693:
18692:
18683:
18682:
18673:
18672:
18670:
18667:
18666:
18647:
18641:
18640:
18639:
18630:
18629:
18627:
18624:
18623:
18602:
18596:
18595:
18594:
18585:
18584:
18582:
18579:
18578:
18561:
18560:
18558:
18555:
18554:
18537:
18531:
18530:
18529:
18527:
18524:
18523:
18502:
18501:
18499:
18496:
18495:
18488:
18480:Gamma functions
18458:
18457:
18455:
18452:
18451:
18412:
18392:
18390:
18372:
18356:
18352:
18317:
18314:
18313:
18294:Godement (1957)
18260:
18255:
18254:
18234:
18216:
18215:
18214:
18210:
18181:
18177:
18157:
18156:
18148:
18143:
18137:
18136:
18134:
18130:
18114:
18109:
18108:
18088:
18070:
18069:
18068:
18064:
18046:
18042:
18036:
18032:
18009:
18008:
18000:
17995:
17989:
17988:
17986:
17982:
17961:
17957:
17940:
17937:
17936:
17909:
17904:
17903:
17883:
17875:
17858:
17837:
17836:
17835:
17831:
17814:
17810:
17790:
17789:
17781:
17776:
17770:
17769:
17767:
17763:
17746:
17743:
17742:
17740:
17672:
17671:
17653:
17649:
17640:
17636:
17631:
17628:
17627:
17586:
17582:
17576:
17572:
17551:
17547:
17541:
17537:
17520:
17517:
17516:
17514:
17507:
17485:
17481:
17475:
17471:
17469:
17466:
17465:
17463:
17432:
17428:
17413:
17409:
17403:
17399:
17397:
17394:
17393:
17365:
17361:
17344:
17340:
17335:
17329:
17325:
17324:
17320:
17289:
17285:
17281:
17279:
17276:
17275:
17223:
17219:
17207:
17203:
17191:
17187:
17185:
17182:
17181:
17161:
17157:
17152:
17146:
17142:
17133:
17129:
17120:
17115:
17109:
17106:
17105:
17075:
17057:
17052:
17046:
17043:
17042:
17013:
17009:
16995:
16992:
16991:
16981:
16977:
16970:
16960:
16903:
16899:
16884:
16880:
16860:
16857:
16856:
16834:
16816:
16811:
16805:
16802:
16801:
16777:
16773:
16768:
16762:
16758:
16749:
16745:
16736:
16731:
16725:
16722:
16721:
16703:
16696:
16689:
16682:
16675:
16668:
16661:
16625:
16621:
16612:
16608:
16603:
16597:
16593:
16584:
16580:
16578:
16575:
16574:
16552:
16548:
16537:
16526:
16523:
16522:
16505:
16504:
16502:
16499:
16498:
16496:
16468:
16464:
16453:
16450:
16449:
16443:
16432:
16425:
16418:
16411:
16404:
16393:
16386:
16380:, intersecting
16367:
16348:
16340:
16313:
16309:
16284:
16268:
16264:
16263:
16261:
16241:
16240:
16234:
16213:
16210:
16209:
16163:
16142:
16138:
16137:
16135:
16133:
16115:
16111:
16088:
16087:
16085:
16082:
16081:
16042:
16037:
16031:
16030:
16027:
16024:
16023:
16017:
15981:symmetric space
15957:
15953:
15938:
15934:
15926:
15923:
15922:
15887:
15886:
15867:
15865:
15863:
15860:
15859:
15827:
15823:
15799:
15798:
15786:
15770:
15766:
15757:
15753:
15744:
15740:
15720:
15717:
15716:
15699:
15698:
15696:
15693:
15692:
15667:
15662:
15656:
15655:
15652:
15649:
15648:
15618:
15613:
15612:
15595:
15580:
15576:
15556:
15555:
15547:
15542:
15536:
15535:
15533:
15529:
15512:
15507:
15506:
15489:
15474:
15470:
15450:
15449:
15441:
15435:
15434:
15433:
15432:
15428:
15420:
15412:
15411:
15406:
15389:
15386:
15385:
15345:
15341:
15323:
15319:
15296:
15295:
15293:
15290:
15289:
15224:
15214:
15210:
15201:
15197:
15196:
15194:
15182:
15178:
15169:
15165:
15163:
15160:
15159:
15140:
15134:
15133:
15132:
15120:
15119:
15117:
15114:
15113:
15089:
15088:
15076:
15075:
15073:
15070:
15069:
15040:
15035:
15034:
15025:
15021:
15010:
15000:
14996:
14983:
14982:
14978:
14970:
14962:
14961:
14956:
14928:
14924:
14922:
14919:
14918:
14880:
14879:
14877:
14874:
14873:
14850:
14835:
14831:
14823:
14820:
14819:
14784:
14780:
14773:
14751:
14747:
14723:
14722:
14710:
14705:
14703:
14691:
14687:
14678:
14674:
14672:
14669:
14668:
14649:
14643:
14642:
14641:
14639:
14636:
14635:
14618:
14617:
14608:
14602:
14601:
14600:
14598:
14595:
14594:
14572:
14568:
14534:
14533:
14512:
14508:
14499:
14495:
14477:
14473:
14464:
14460:
14458:
14455:
14454:
14431:
14425:
14424:
14423:
14421:
14418:
14417:
14411:
14364:
14361:
14360:
14332:
14331:
14306:
14305:
14288:
14285:
14284:
14262:
14261:
14259:
14256:
14255:
14223:
14222:
14210:
14209:
14200:
14199:
14197:
14194:
14193:
14176:
14175:
14173:
14170:
14169:
14152:
14151:
14149:
14146:
14145:
14130:
14123:
14104:
14103:
14101:
14098:
14097:
14058:
14016:
14014:
14010:
13991:
13989:
13974:
13970:
13950:
13949:
13943:
13938:
13926:
13922:
13917:
13900:
13897:
13896:
13892:
13844:
13840:
13817:
13816:
13798:
13787:
13786:
13785:
13783:
13780:
13779:
13758:
13754:
13734:
13733:
13724:
13720:
13705:
13701:
13699:
13696:
13695:
13675:
13653:
13647:
13646:
13645:
13643:
13640:
13639:
13592:
13588:
13570:
13566:
13564:
13561:
13560:
13506:
13504:
13500:
13481:
13479:
13459:
13458:
13452:
13444:
13431:
13427:
13423:
13418:
13398:
13379:
13370:
13357:
13355:
13349:
13341:
13307:
13306:
13300:
13295:
13282:
13278:
13274:
13269:
13252:
13249:
13248:
13182:
13181:
13175:
13170:
13156:
13133:
13130:
13129:
13101:
13100:
13098:
13095:
13094:
13027:
13023:
13021:
13014:
13010:
13001:
12995:
12990:
12978:
12944:
12940:
12931:
12927:
12926:
12924:
12917:
12913:
12904:
12898:
12890:
12880:
12872:
12842:
12838:
12836:
12829:
12825:
12816:
12810:
12802:
12796:
12793:
12792:
12751:
12747:
12745:
12738:
12734:
12725:
12719:
12711:
12697:
12689:
12687:
12670:
12667:
12666:
12614:
12610:
12586:
12581:
12555:
12551:
12532:
12528:
12519:
12515:
12514:
12512:
12508:
12498:
12493:
12466:
12465:
12463:
12460:
12459:
12418:
12414:
12412:
12405:
12401:
12391:
12383:
12362:
12359:
12358:
12312:
12302:
12298:
12282:
12278:
12266:
12262:
12253:
12249:
12248:
12246:
12242:
12232:
12227:
12217:
12209:
12182:
12181:
12179:
12176:
12175:
12135:
12117:
12113:
12090:
12089:
12087:
12084:
12083:
12042:
12036:
12032:
12014:
12010:
12008:
12005:
12004:
12002:
11997:
11955:
11953:
11949:
11930:
11928:
11907:
11906:
11891:
11887:
11881:
11873:
11852:
11849:
11848:
11789:
11788:
11770:
11757:
11753:
11749:
11748:
11746:
11743:
11742:
11720:
11716:
11701:
11697:
11695:
11692:
11691:
11644:
11640:
11634:
11630:
11613:
11610:
11609:
11592:
11588:
11564:
11560:
11554:
11550:
11548:
11545:
11544:
11520:
11516:
11510:
11506:
11494:
11490:
11484:
11480:
11478:
11475:
11474:
11422:
11418:
11417:
11413:
11377:
11373:
11372:
11368:
11366:
11363:
11362:
11336:
11332:
11290:
11286:
11284:
11281:
11280:
11249:
11245:
11237:
11233:
11232:
11228:
11207:
11203:
11202:
11198:
11196:
11193:
11192:
11165:
11132:
11128:
11126:
11123:
11122:
11088:
11044:
11040:
11036:
11030:
11026:
11017:
11013:
10996:
10992:
10991:
10987:
10966:
10962:
10961:
10957:
10955:
10952:
10951:
10923:
10919:
10905:
10901:
10900:
10896:
10885:
10882:
10881:
10876:, the function
10867:
10844:
10829:
10799:
10795:
10789:
10785:
10770:
10766:
10760:
10756:
10754:
10751:
10750:
10744:
10712:
10708:
10684:
10680:
10671:
10667:
10662:
10659:
10658:
10620:
10616:
10586:
10581:
10568:
10564:
10528:
10523:
10517:
10514:
10513:
10488:
10484:
10445:
10440:
10424:
10420:
10405:
10401:
10380:
10375:
10356:
10352:
10343:
10339:
10331:
10328:
10327:
10280:
10276:
10267:
10263:
10242:
10238:
10236:
10233:
10232:
10163:
10160:
10159:
10087:
10079:
10048:
10044:
10043:
10039:
10014:
10010:
9992:
9984:
9953:
9949:
9948:
9944:
9942:
9939:
9938:
9893:
9889:
9877:
9873:
9861:
9857:
9845:
9841:
9829:
9824:
9811:
9806:
9793:
9788:
9775:
9771:
9769:
9766:
9765:
9744:
9740:
9728:
9724:
9712:
9708:
9696:
9691:
9678:
9673:
9660:
9656:
9654:
9651:
9650:
9625:
9621:
9597:
9592:
9579:
9575:
9563:
9559:
9557:
9554:
9553:
9516:
9513:
9512:
9444:
9437:
9433:
9427:
9423:
9368:
9361:
9357:
9351:
9347:
9327:
9324:
9323:
9251:
9227:
9223:
9214:
9210:
9201:
9197:
9182:
9178:
9176:
9173:
9172:
9146:
9145:
9104:
9053:
9052:
9002:
8956:
8955:
8940:
8936:
8934:
8931:
8930:
8908:
8904:
8884:
8881:
8880:
8862:
8861:
8850:
8835:
8834:
8820:
8804:
8803:
8795:
8792:
8791:
8756:
8755:
8744:
8726:
8725:
8711:
8695:
8694:
8685:
8681:
8679:
8676:
8675:
8671:
8645:
8639:
8638:
8637:
8635:
8632:
8631:
8624:
8616:direct integral
8595:
8589:
8588:
8587:
8578:
8574:
8572:
8569:
8568:
8533:
8529:
8514:
8510:
8488:
8484:
8480:
8454:
8451:
8450:
8413:
8411:
8407:
8388:
8386:
8384:
8381:
8380:
8361:
8334:
8330:
8318:
8312:
8311:
8310:
8301:
8297:
8289:
8286:
8285:
8283:
8256:
8252:
8243:
8239:
8224:
8220:
8199:
8195:
8193:
8190:
8189:
8165:
8154:
8149:
8148:
8140:
8133:
8129:
8128:
8119:
8115:
8097:
8093:
8091:
8088:
8087:
8030:
8025:
8024:
8007:
7980:
7976:
7967:
7963:
7961:
7958:
7957:
7938:
7932:
7931:
7930:
7928:
7925:
7924:
7914:
7883:
7882:
7873:
7869:
7860:
7855:
7846:
7843:
7842:
7815:
7804:
7800:
7796:
7785:
7768:
7766:
7763:
7762:
7745:compact support
7693:
7689:
7671:
7667:
7649:
7645:
7639:
7635:
7625:
7621:
7615:
7610:
7606:
7602:
7598:
7589:
7588:
7586:
7583:
7582:
7547:
7543:
7526:
7517:
7513:
7512:
7508:
7499:
7495:
7489:
7485:
7483:
7480:
7479:
7471:
7429:
7427:
7423:
7404:
7402:
7386:
7382:
7380:
7378:
7372:
7361:
7360:
7359:
7353:
7345:
7320:
7316:
7314:
7308:
7304:
7296:
7290:
7289:
7288:
7287:
7283:
7281:
7278:
7277:
7269:
7265:
7236:
7230:
7229:
7228:
7226:
7223:
7222:
7182:
7180:
7176:
7157:
7155:
7135:
7134:
7119:
7115:
7109:
7101:
7080:
7077:
7076:
7060:
7015:
7014:
6996:
6983:
6978:
6970:
6969:
6967:
6964:
6963:
6929:
6925:
6896:
6895:
6893:
6890:
6889:
6870:
6864:
6863:
6862:
6860:
6857:
6856:
6834:
6830:
6824:
6820:
6811:
6807:
6805:
6802:
6801:
6758:
6722:
6718:
6717:
6708:
6703:
6689:
6684:
6669:
6665:
6656:
6652:
6650:
6647:
6646:
6626:
6605:
6601:
6595:
6591:
6582:
6578:
6560:
6556:
6554:
6551:
6550:
6511:
6499:
6495:
6493:
6490:
6489:
6460:
6458:
6442:
6416:
6402:
6400:
6379:
6375:
6358:
6355:
6354:
6332:
6328:
6311:
6302:
6298:
6297:
6293:
6284:
6280:
6274:
6270:
6268:
6265:
6264:
6247:
6243:
6237:
6232:
6219:
6215:
6213:
6210:
6209:
6187:
6182:
6161:
6152:
6148:
6147:
6143:
6134:
6130:
6124:
6119:
6113:
6110:
6109:
6071:
6066:
6045:
6039:
6038:
6037:
6036:
6032:
6004:
6000:
5989:
5983:
5982:
5981:
5980:
5976:
5974:
5971:
5970:
5960:
5920:
5915:
5894:
5888:
5887:
5886:
5885:
5881:
5853:
5849:
5838:
5832:
5831:
5830:
5829:
5825:
5823:
5820:
5819:
5767:
5759:
5745:
5741:
5737:
5710:
5705:
5699:
5696:
5695:
5691:
5653:
5644:
5640:
5639:
5635:
5629:
5625:
5616:
5612:
5606:
5602:
5600:
5597:
5596:
5589:
5582:
5576:, the operator
5575:
5549:
5543:
5542:
5541:
5539:
5536:
5535:
5533:
5493:
5484:
5480:
5479:
5475:
5469:
5465:
5453:
5449:
5443:
5439:
5437:
5434:
5433:
5387:
5383:
5379:
5346:
5342:
5340:
5337:
5336:
5317:
5311:
5310:
5309:
5307:
5304:
5303:
5288:
5260:applied to the
5216:
5212:
5194:
5186:
5164:
5161:
5160:
5138:
5134:
5116:
5111:
5096:
5093:
5092:
5052:
5046:
5045:
5044:
5042:
5039:
5038:
5037:Every point in
5013:
5008:
4995:
4990:
4977:
4973:
4962:
4959:
4958:
4917:
4913:
4902:
4899:
4898:
4872:
4868:
4856:
4852:
4848:
4844:
4835:
4831:
4822:
4818:
4813:
4810:
4809:
4775:
4769:
4768:
4767:
4765:
4762:
4761:
4728:
4724:
4674:
4671:
4670:
4648:
4647:
4642:
4636:
4635:
4630:
4620:
4619:
4611:
4608:
4607:
4549:
4543:
4542:
4541:
4539:
4536:
4535:
4522:
4514:direct integral
4493:
4487:
4486:
4485:
4476:
4472:
4470:
4467:
4466:
4431:
4427:
4412:
4408:
4386:
4382:
4378:
4352:
4349:
4348:
4331:
4311:
4305:
4304:
4303:
4294:
4290:
4288:
4285:
4284:
4278:
4251:
4247:
4238:
4234:
4219:
4215:
4194:
4190:
4188:
4185:
4184:
4160:
4149:
4144:
4143:
4135:
4128:
4124:
4123:
4114:
4110:
4092:
4088:
4086:
4083:
4082:
4025:
4020:
4019:
4002:
3975:
3971:
3962:
3958:
3956:
3953:
3952:
3933:
3927:
3926:
3925:
3923:
3920:
3919:
3909:
3870:
3866:
3862:
3858:
3847:
3830:
3828:
3825:
3824:
3807:compact support
3733:
3729:
3711:
3707:
3701:
3697:
3687:
3683:
3677:
3672:
3668:
3664:
3660:
3651:
3650:
3648:
3645:
3644:
3609:
3605:
3590:
3586:
3574:
3570:
3565:
3562:
3561:
3553:
3526:
3525:
3523:
3520:
3519:
3497:
3479:
3475:
3467:
3464:
3463:
3444:
3443:
3432:
3431:
3416:
3415:
3399:
3395:
3387:
3378:
3374:
3360:
3342:
3338:
3325:
3324:
3322:
3319:
3318:
3283:
3279:
3257:
3246:
3245:
3244:
3243:
3241:
3226:
3215:
3214:
3213:
3189:
3185:
3183:
3177:
3173:
3167:
3163:
3161:
3158:
3157:
3119:
3115:
3108:
3106:
3088:
3084:
3082:
3079:
3078:
3068:
3064:
3040:
3036:
3027:
3022:
3010:
3007:
3006:
2977:
2973:
2958:
2954:
2934:
2933:
2913:
2910:
2909:
2878:
2874:
2845:
2844:
2842:
2839:
2838:
2804:
2790:
2788:
2770:
2766:
2764:
2761:
2760:
2714:
2710:
2706:
2700:
2696:
2694:
2676:
2672:
2670:
2667:
2666:
2605:
2601:
2582:
2574:
2552:
2549:
2548:
2526:
2522:
2501:
2496:
2481:
2478:
2477:
2437:
2431:
2430:
2429:
2427:
2424:
2423:
2422:Every point in
2401:
2397:
2382:
2377:
2364:
2359:
2346:
2341:
2328:
2324:
2313:
2310:
2309:
2261:
2257:
2246:
2243:
2242:
2216:
2212:
2200:
2196:
2184:
2180:
2176:
2172:
2163:
2159:
2150:
2146:
2141:
2138:
2137:
2127:It carries the
2106:
2094:
2088:
2087:
2086:
2084:
2081:
2080:
2047:
2043:
1993:
1990:
1989:
1971:
1970:
1965:
1959:
1958:
1953:
1943:
1942:
1934:
1931:
1930:
1867:
1861:
1860:
1859:
1857:
1854:
1853:
1838:
1827:
1823:
1756:
1752:
1750:
1747:
1746:
1699:
1694:
1693:
1676:
1670:
1666:
1657:
1653:
1645:
1642:
1641:
1612:
1608:
1573:
1570:
1569:
1516:
1512:
1508:
1479:
1476:
1475:
1442:
1414:
1409:
1403:
1327:
1323:
1318:
1303:
1299:
1291:
1288:
1287:
1270:
1266:
1262:
1258:
1243:
1190:
1186:
1182:
1167:
1163:
1140:
1138:
1135:
1134:
1033:
1029:
1014:
1010:
1004:
1000:
982:
978:
976:
973:
972:
955:
925:
924:
922:
919:
918:
900:
855:
851:
830:
826:
799:
795:
793:
790:
789:
783:
753:
752:
750:
747:
746:
744:
713:
666:
665:
663:
660:
659:
649:
633:locally compact
625:spectral theory
622:
603:
581:
580:
578:
575:
574:
545:
539:
538:
537:
535:
532:
531:
495:
494:
492:
489:
488:
463:
462:
460:
457:
456:
428:
427:
425:
422:
421:
411:symmetric space
361:
327:
321:
309:Godement (1957)
239:
211:Euclidean space
208:
135:
127:Helgason (1984)
92:
74:symmetric space
60:of the general
12:
11:
5:
27376:
27366:
27365:
27360:
27355:
27338:
27337:
27335:
27334:
27323:
27320:
27319:
27317:
27316:
27311:
27306:
27301:
27299:Choquet theory
27296:
27291:
27285:
27283:
27279:
27278:
27276:
27275:
27265:
27260:
27255:
27250:
27245:
27240:
27235:
27230:
27225:
27220:
27215:
27209:
27207:
27203:
27202:
27200:
27199:
27194:
27188:
27186:
27182:
27181:
27179:
27178:
27173:
27168:
27163:
27158:
27153:
27151:Banach algebra
27147:
27145:
27141:
27140:
27138:
27137:
27132:
27127:
27122:
27117:
27112:
27107:
27102:
27097:
27092:
27086:
27084:
27080:
27079:
27077:
27076:
27074:Banach–Alaoglu
27071:
27066:
27061:
27056:
27051:
27046:
27041:
27036:
27030:
27028:
27022:
27021:
27018:
27017:
27015:
27014:
27009:
27004:
27002:Locally convex
26999:
26985:
26980:
26974:
26972:
26968:
26967:
26965:
26964:
26959:
26954:
26949:
26944:
26939:
26934:
26929:
26924:
26919:
26913:
26907:
26903:
26902:
26886:
26885:
26878:
26871:
26863:
26857:
26856:
26834:
26823:(2): 691–707,
26810:
26794:
26775:(1): 143–149,
26759:
26750:
26745:
26729:
26724:
26708:
26697:(4): 627–642,
26676:
26671:
26655:
26650:
26633:
26608:
26607:
26601:
26588:
26583:
26566:
26556:
26529:
26494:
26464:
26426:
26389:(4): 337–342,
26378:
26323:Harish-Chandra
26319:
26309:
26289:
26273:
26256:(4): 290–297,
26245:
26213:
26199:Naimark, M. A.
26195:Gelfand, I. M.
26191:
26186:
26173:
26153:
26148:
26132:
26127:
26111:
26087:
26059:
26048:(2): 331–349,
26035:
26032:
26030:
26029:
26017:
26005:
25993:
25990:Rosenberg 1977
25982:
25970:
25957:
25945:
25933:
25921:
25909:
25897:
25885:
25873:
25861:
25849:
25837:
25825:
25813:
25801:
25789:
25777:
25766:
25745:
25726:
25720:
25714:
25703:
25689:
25677:
25665:
25653:
25634:
25630:
25626:
25623:
25619:
25615:
25612:
25609:
25604:
25600:
25579:
25568:
25556:
25544:
25535:
25524:
25521:Takahashi 1963
25513:
25501:
25489:
25478:
25467:
25451:
25439:
25428:
25417:
25400:
25385:
25359:
25347:
25336:
25322:
25306:
25295:
25284:
25273:
25262:
25249:
25247:
25244:
25220:
25215:
25209:
25178:
25173:
25169:
25163:
25157:
25151:
25146:
25122:
25118:
25114:
25111:
25107:
25103:
25100:
25097:
25094:
25089:
25085:
25081:
25078:
25075:
25072:
25067:
25063:
25059:
25056:
25053:
25050:
25047:
25044:
25041:
25038:
25035:
25031:
25025:
25021:
25017:
25012:
25008:
25004:
25000:
24996:
24993:
24988:
24984:
24980:
24977:
24973:
24969:
24966:
24963:
24959:
24955:
24952:
24949:
24946:
24941:
24925:
24922:
24907:
24875:
24872:
24869:
24863:
24860:
24855:
24850:
24847:
24844:
24841:
24837:
24831:
24827:
24823:
24820:
24817:
24811:
24808:
24798:
24793:
24787:
24780:
24776:
24773:
24770:
24750:
24747:
24744:
24740:
24737:
24734:
24729:
24726:
24722:
24718:
24715:
24710:
24706:
24702:
24699:
24696:
24693:
24688:
24684:
24661:
24658:
24655:
24652:
24649:
24646:
24643:
24640:
24637:
24634:
24631:
24595:
24592:
24586:
24583:
24578:
24573:
24570:
24567:
24564:
24560:
24556:
24553:
24550:
24544:
24541:
24531:
24526:
24520:
24513:
24509:
24506:
24503:
24500:
24497:
24466:
24448:
24443:
24437:
24414:
24411:
24408:
24405:
24400:
24394:
24369:
24363:
24330:
24327:
24282:
24276:
24272:
24269:
24266:
24262:
24258:
24254:
24248:
24245:
24241:
24236:
24232:
24228:
24224:
24221:
24218:
24213:
24209:
24205:
24201:
24197:
24194:
24191:
24185:
24182:
24175:
24136:
24133:
24126:
24101:
24097:
24093:
24090:
24087:
24082:
24079:
24075:
24071:
24066:
24062:
24058:
24052:
24049:
24042:
24037:
24030:
24027:
24023:
24019:
24016:
24012:
24006:
24002:
23997:
23993:
23988:
23984:
23963:
23956:
23951:
23947:
23942:
23938:
23934:
23931:
23928:
23925:
23922:
23917:
23914:
23910:
23906:
23901:
23897:
23891:
23888:
23882:
23876:
23873:
23867:
23862:
23858:
23853:
23849:
23844:
23840:
23836:
23833:
23830:
23827:
23824:
23821:
23818:
23813:
23809:
23803:
23800:
23794:
23788:
23785:
23779:
23774:
23769:
23766:
23761:
23757:
23753:
23750:
23747:
23744:
23741:
23738:
23733:
23728:
23724:
23720:
23717:
23714:
23711:
23708:
23704:
23695:
23691:
23687:
23684:
23681:
23678:
23671:
23667:
23662:
23650:
23649:
23646:beta functions
23642:
23626:
23604:
23589:
23575:
23572:
23566:
23543:
23530:
23513:
23510:
23505:
23501:
23497:
23494:
23491:
23488:
23483:
23479:
23475:
23472:
23469:
23466:
23461:
23457:
23451:
23447:
23443:
23440:
23420:
23417:
23414:
23407:
23403:
23398:
23394:
23391:
23386:
23382:
23378:
23371:
23367:
23362:
23358:
23355:
23352:
23349:
23342:
23338:
23332:
23328:
23323:
23294:
23282:
23277:
23273:
23269:
23266:
23263:
23258:
23254:
23250:
23245:
23241:
23237:
23234:
23231:
23228:
23225:
23222:
23196:
23193:
23190:
23187:
23180:
23176:
23171:
23167:
23164:
23161:
23158:
23155:
23127:
23122:
23116:
23110:
23088:
23083:
23077:
23062:
23054:for any point
23025:
23020:
23016:
23012:
23009:
23006:
23003:
22998:
22994:
22990:
22987:
22984:
22981:
22976:
22972:
22966:
22962:
22958:
22955:
22935:
22932:
22929:
22926:
22921:
22917:
22913:
22910:
22907:
22904:
22899:
22895:
22891:
22886:
22882:
22878:
22875:
22872:
22869:
22866:
22863:
22858:
22854:
22848:
22844:
22840:
22837:
22815:
22812:
22809:
22806:
22803:
22800:
22797:
22794:
22791:
22788:
22783:
22780:
22776:
22772:
22769:
22766:
22763:
22758:
22754:
22750:
22747:
22744:
22741:
22738:
22735:
22715:
22712:
22709:
22705:
22702:
22699:
22696:
22693:
22690:
22685:
22682:
22677:
22674:
22670:
22666:
22663:
22660:
22657:
22654:
22650:
22646:
22643:
22640:
22637:
22634:
22631:
22628:
22625:
22622:
22619:
22616:
22562:
22556:
22507:
22496:
22484:
22474:
22456:
22452:
22429:Main article:
22426:
22423:
22411:
22408:
22405:
22399:
22395:
22392:
22389:
22385:
22382:
22373:
22370:
22367:
22364:
22360:
22356:
22353:
22350:
22347:
22344:
22332:
22318:
22312:
22283:
22280:
22275:
22272:
22269:
22265:
22261:
22256:
22253:
22249:
22243:
22240:
22237:
22233:
22207:
22181:
22178:
22175:
22168:
22163:
22160:
22156:
22152:
22147:
22143:
22139:
22135:
22132:
22125:
22121:
22118:
22115:
22111:
22108:
22097:
22094:
22091:
22088:
22084:
22078:
22075:
22072:
22069:
22065:
22061:
22058:
22053:
22049:
22045:
22040:
22036:
22009:
22006:
22003:
21997:
21994:
21990:
21986:
21983:
21980:
21976:
21973:
21967:
21963:
21959:
21955:
21951:
21948:
21945:
21942:
21937:
21933:
21911:
21877:
21871:
21865:
21843:
21837:
21818:
21790:
21787:
21784:
21781:
21776:
21773:
21770:
21766:
21762:
21759:
21742:(λ) using the
21735:
21731:> 0 large.
21714:
21708:
21679:
21676:
21673:
21670:
21667:
21664:
21661:
21658:
21654:
21650:
21647:
21644:
21641:
21638:
21635:
21632:
21627:
21623:
21619:
21614:
21610:
21587:
21581:
21566:
21544:
21539:
21536:
21532:
21528:
21525:
21522:
21519:
21516:
21511:
21508:
21505:
21501:
21497:
21492:
21488:
21471:
21464:
21447:
21417:
21392:
21386:
21371:
21354:
21349:
21346:
21343:
21340:
21336:
21330:
21327:
21324:
21320:
21316:
21313:
21310:
21307:
21304:
21301:
21298:
21282:
21267:
21262:
21259:
21255:
21251:
21248:
21245:
21240:
21236:
21230:
21227:
21224:
21220:
21214:
21211:
21208:
21205:
21201:
21197:
21192:
21188:
21175:
21171:
21143:
21111:
21107:
21102:
21096:
21091:
21088:
21085:
21080:
21075:
21070:
21067:
21064:
21058:
21054:
21049:
21045:
21041:
21029:
21022:
21003:
20998:
20994:
20988:
20985:
20982:
20978:
20974:
20968:
20963:
20959:
20954:
20950:
20929:
20924:
20920:
20916:
20913:
20910:
20907:
20904:
20901:
20898:
20892:
20888:
20882:
20879:
20876:
20872:
20868:
20863:
20859:
20855:
20852:
20835:
20818:
20794:
20789:
20785:
20781:
20778:
20775:
20769:
20764:
20743:
20740:
20737:
20734:
20731:
20728:
20725:
20722:
20717:
20713:
20709:
20700:is defined by
20687:
20673:
20658:
20653:
20649:
20644:
20641:
20638:
20632:
20628:
20622:
20619:
20616:
20612:
20608:
20602:
20597:
20593:
20590:
20565:
20560:
20557:
20554:
20549:
20545:
20542:
20539:
20536:
20533:
20530:
20527:
20521:
20518:
20514:
20508:
20504:
20501:
20498:
20495:
20492:
20489:
20467:
20431:
20387:
20349:
20346:
20330:
20327:
20304:
20301:
20297:
20293:
20290:
20287:
20284:
20279:
20275:
20271:
20268:
20248:
20245:
20242:
20236:
20233:
20228:
20223:
20220:
20217:
20214:
20210:
20206:
20201:
20196:
20190:
20184:
20179:
20175:
20171:
20168:
20165:
20161:
20157:
20154:
20151:
20148:
20143:
20139:
20129:
20123:
20120:
20114:
20111:
20108:
20105:
20079:
20076:
20071:
20067:
20059:
20055:
20051:
20048:
20044:
20038:
20035:
20030:
20027:
20007:
19992:
19987:
19984:
19980:
19976:
19973:
19970:
19967:
19964:
19959:
19955:
19930:
19925:
19921:
19916:
19912:
19908:
19905:
19902:
19899:
19896:
19891:
19888:
19884:
19880:
19875:
19871:
19865:
19862:
19856:
19850:
19847:
19841:
19836:
19832:
19827:
19823:
19818:
19814:
19810:
19807:
19804:
19801:
19798:
19795:
19792:
19787:
19783:
19777:
19774:
19768:
19762:
19759:
19753:
19748:
19743:
19740:
19735:
19731:
19727:
19724:
19721:
19718:
19715:
19712:
19707:
19702:
19698:
19694:
19691:
19688:
19685:
19682:
19678:
19667:
19662:
19658:
19654:
19651:
19647:
19643:
19638:
19634:
19630:
19627:
19624:
19621:
19618:
19591:
19588:
19585:
19579:
19576:
19571:
19566:
19563:
19560:
19557:
19553:
19548:
19545:
19542:
19537:
19533:
19529:
19526:
19523:
19517:
19514:
19504:
19499:
19493:
19486:
19482:
19479:
19476:
19473:
19470:
19444:
19441:
19438:
19434:
19431:
19428:
19423:
19420:
19416:
19412:
19409:
19406:
19403:
19398:
19394:
19390:
19387:
19384:
19381:
19375:
19372:
19345:
19327:
19321:
19306:
19299:
19281:
19276:
19270:
19247:
19244:
19241:
19236:
19232:
19227:
19223:
19220:
19217:
19212:
19208:
19204:
19201:
19171:
19147:
19141:
19116:
19092:
19056:
19037:is called its
19024:
19018:
19012:
19009:
19006:
19001:
18997:
18972:
18969:
18966:
18963:
18958:
18952:
18927:
18924:
18921:
18916:
18911:
18908:
18905:
18900:
18897:
18894:
18891:
18888:
18885:
18882:
18879:
18876:
18873:
18870:
18867:
18862:
18857:
18854:
18851:
18848:
18843:
18837:
18810:
18804:
18779:
18755:
18731:
18728:
18725:
18720:
18715:
18712:
18709:
18706:
18701:
18696:
18691:
18686:
18681:
18676:
18650:
18644:
18638:
18633:
18605:
18599:
18593:
18588:
18564:
18540:
18534:
18505:
18487:
18484:
18461:
18433:
18427:
18424:
18421:
18418:
18415:
18410:
18407:
18404:
18401:
18398:
18395:
18389:
18386:
18381:
18378:
18375:
18371:
18367:
18362:
18359:
18355:
18351:
18348:
18345:
18342:
18339:
18336:
18333:
18330:
18327:
18324:
18321:
18275:
18272:
18269:
18263:
18258:
18253:
18250:
18247:
18244:
18241:
18237:
18231:
18225:
18222:
18219:
18213:
18209:
18206:
18203:
18200:
18195:
18192:
18189:
18184:
18180:
18176:
18173:
18170:
18164:
18161:
18151:
18146:
18140:
18133:
18129:
18126:
18123:
18117:
18112:
18107:
18104:
18101:
18098:
18095:
18091:
18085:
18079:
18076:
18073:
18067:
18063:
18060:
18057:
18052:
18049:
18045:
18039:
18035:
18031:
18028:
18025:
18022:
18016:
18013:
18003:
17998:
17992:
17985:
17981:
17978:
17975:
17972:
17969:
17964:
17960:
17956:
17953:
17950:
17947:
17944:
17924:
17921:
17918:
17912:
17907:
17902:
17899:
17896:
17893:
17890:
17886:
17882:
17878:
17874:
17871:
17868:
17865:
17861:
17857:
17852:
17846:
17843:
17840:
17834:
17828:
17825:
17822:
17817:
17813:
17809:
17806:
17803:
17797:
17794:
17784:
17779:
17773:
17766:
17762:
17759:
17756:
17753:
17750:
17738:
17697:
17694:
17691:
17688:
17685:
17679:
17676:
17670:
17667:
17664:
17661:
17656:
17652:
17648:
17643:
17639:
17635:
17613:
17610:
17607:
17603:
17600:
17597:
17592:
17589:
17585:
17579:
17575:
17571:
17568:
17565:
17562:
17557:
17554:
17550:
17544:
17540:
17536:
17533:
17530:
17527:
17524:
17512:
17505:
17491:
17488:
17484:
17478:
17474:
17461:
17440:
17435:
17431:
17427:
17424:
17421:
17416:
17412:
17406:
17402:
17392:In particular
17379:
17376:
17373:
17368:
17364:
17360:
17357:
17354:
17347:
17343:
17338:
17332:
17328:
17323:
17319:
17316:
17313:
17310:
17307:
17304:
17301:
17296:
17292:
17288:
17284:
17253:
17250:
17247:
17243:
17240:
17237:
17234:
17231:
17226:
17222:
17218:
17215:
17210:
17206:
17202:
17199:
17194:
17190:
17169:
17164:
17160:
17155:
17149:
17145:
17141:
17136:
17132:
17128:
17123:
17118:
17114:
17085:
17082:
17078:
17074:
17071:
17068:
17065:
17060:
17055:
17051:
17024:
17021:
17016:
17012:
17008:
17005:
17002:
16999:
16979:
16975:
16968:
16958:
16921:
16918:
16915:
16911:
16906:
16902:
16898:
16895:
16892:
16887:
16883:
16879:
16876:
16873:
16870:
16867:
16864:
16844:
16841:
16837:
16833:
16830:
16827:
16824:
16819:
16814:
16810:
16785:
16780:
16776:
16771:
16765:
16761:
16757:
16752:
16748:
16744:
16739:
16734:
16730:
16701:
16694:
16687:
16680:
16673:
16666:
16659:
16628:
16624:
16620:
16615:
16611:
16606:
16600:
16596:
16592:
16587:
16583:
16560:
16555:
16551:
16547:
16544:
16540:
16536:
16533:
16530:
16508:
16494:
16479:
16476:
16471:
16467:
16463:
16460:
16457:
16441:
16430:
16423:
16416:
16409:
16402:
16391:
16384:
16365:
16346:
16339:
16336:
16321:
16316:
16312:
16308:
16305:
16302:
16296:
16293:
16290:
16287:
16282:
16279:
16276:
16271:
16267:
16260:
16257:
16254:
16248:
16245:
16237:
16233:
16229:
16226:
16223:
16220:
16217:
16200:Fourier series
16185:
16182:
16175:
16172:
16169:
16166:
16160:
16156:
16153:
16150:
16145:
16141:
16132:
16129:
16126:
16123:
16118:
16114:
16110:
16107:
16104:
16101:
16095:
16092:
16072:class function
16045:
16040:
16034:
16015:
15979:Note that the
15965:
15960:
15956:
15952:
15949:
15946:
15941:
15937:
15933:
15930:
15906:
15903:
15900:
15894:
15891:
15883:
15879:
15876:
15873:
15870:
15847:
15842:
15839:
15836:
15833:
15830:
15826:
15822:
15819:
15816:
15811:
15808:
15805:
15802:
15795:
15792:
15789:
15785:
15781:
15778:
15773:
15769:
15765:
15760:
15756:
15752:
15747:
15743:
15739:
15736:
15733:
15730:
15727:
15724:
15702:
15670:
15665:
15659:
15634:
15631:
15627:
15621:
15616:
15611:
15608:
15605:
15602:
15598:
15594:
15591:
15588:
15583:
15579:
15575:
15572:
15569:
15563:
15560:
15550:
15545:
15539:
15532:
15528:
15525:
15521:
15515:
15510:
15505:
15502:
15499:
15496:
15492:
15488:
15485:
15482:
15477:
15473:
15469:
15466:
15463:
15457:
15454:
15444:
15438:
15431:
15423:
15419:
15415:
15410:
15405:
15402:
15399:
15396:
15393:
15369:
15366:
15362:
15359:
15356:
15351:
15348:
15344:
15340:
15337:
15334:
15331:
15326:
15322:
15318:
15315:
15312:
15309:
15303:
15300:
15261:Borel subgroup
15242:
15236:
15233:
15230:
15227:
15222:
15217:
15213:
15209:
15204:
15200:
15193:
15190:
15185:
15181:
15177:
15172:
15168:
15143:
15137:
15131:
15128:
15123:
15092:
15087:
15084:
15079:
15055:
15052:
15049:
15043:
15038:
15033:
15028:
15024:
15020:
15017:
15013:
15008:
15003:
14999:
14995:
14992:
14986:
14981:
14973:
14969:
14965:
14960:
14955:
14952:
14949:
14945:
14942:
14939:
14936:
14931:
14927:
14883:
14857:
14853:
14849:
14846:
14843:
14838:
14834:
14830:
14827:
14798:
14792:
14787:
14783:
14779:
14776:
14769:
14766:
14763:
14760:
14757:
14754:
14750:
14746:
14743:
14740:
14735:
14732:
14729:
14726:
14719:
14716:
14713:
14709:
14702:
14699:
14694:
14690:
14686:
14681:
14677:
14652:
14646:
14621:
14616:
14611:
14605:
14580:
14575:
14571:
14567:
14564:
14561:
14558:
14555:
14552:
14549:
14545:
14542:
14537:
14532:
14529:
14526:
14523:
14520:
14515:
14511:
14507:
14502:
14498:
14494:
14491:
14488:
14485:
14480:
14476:
14472:
14467:
14463:
14434:
14428:
14409:
14395:
14392:
14389:
14386:
14383:
14380:
14377:
14374:
14371:
14368:
14340:
14335:
14330:
14327:
14324:
14321:
14318:
14314:
14309:
14304:
14301:
14298:
14295:
14292:
14270:
14265:
14231:
14226:
14221:
14218:
14213:
14208:
14203:
14179:
14155:
14129:
14126:
14121:
14107:
14062:Lorentz groups
14057:
14054:
14042:
14039:
14036:
14031:
14026:
14022:
14019:
14013:
14009:
14006:
14001:
13997:
13994:
13988:
13985:
13982:
13977:
13973:
13969:
13966:
13963:
13957:
13954:
13946:
13941:
13937:
13929:
13925:
13921:
13916:
13913:
13910:
13907:
13904:
13890:
13861:
13858:
13855:
13852:
13847:
13843:
13839:
13836:
13833:
13830:
13824:
13821:
13815:
13812:
13809:
13806:
13801:
13794:
13791:
13761:
13757:
13753:
13750:
13747:
13741:
13738:
13732:
13727:
13723:
13719:
13716:
13713:
13708:
13704:
13673:
13656:
13650:
13625:
13622:
13619:
13615:
13612:
13609:
13606:
13603:
13600:
13595:
13591:
13587:
13584:
13581:
13578:
13573:
13569:
13532:
13529:
13526:
13521:
13516:
13512:
13509:
13503:
13499:
13496:
13491:
13487:
13484:
13478:
13475:
13472:
13466:
13463:
13455:
13450:
13447:
13443:
13434:
13430:
13426:
13422:
13417:
13414:
13411:
13405:
13402:
13397:
13394:
13388:
13385:
13382:
13377:
13373:
13369:
13366:
13363:
13360:
13352:
13347:
13344:
13340:
13336:
13333:
13329:
13326:
13323:
13320:
13314:
13311:
13303:
13298:
13294:
13285:
13281:
13277:
13273:
13268:
13265:
13262:
13259:
13256:
13232:
13229:
13226:
13222:
13219:
13216:
13213:
13210:
13207:
13204:
13201:
13198:
13195:
13189:
13186:
13178:
13173:
13169:
13163:
13160:
13155:
13152:
13149:
13146:
13143:
13140:
13137:
13108:
13105:
13076:
13073:
13070:
13067:
13064:
13061:
13058:
13055:
13052:
13049:
13045:
13041:
13035:
13030:
13026:
13020:
13017:
13013:
13007:
13004:
12998:
12993:
12989:
12984:
12981:
12977:
12974:
12971:
12967:
12964:
12959:
12953:
12947:
12943:
12939:
12934:
12930:
12923:
12920:
12916:
12910:
12907:
12901:
12896:
12893:
12889:
12883:
12878:
12875:
12871:
12867:
12864:
12861:
12856:
12850:
12845:
12841:
12835:
12832:
12828:
12822:
12819:
12813:
12808:
12805:
12801:
12776:
12773:
12770:
12765:
12759:
12754:
12750:
12744:
12741:
12737:
12731:
12728:
12722:
12717:
12714:
12710:
12703:
12700:
12695:
12692:
12686:
12683:
12680:
12677:
12674:
12638:
12635:
12632:
12626:
12623:
12620:
12617:
12613:
12609:
12606:
12603:
12600:
12597:
12594:
12589:
12584:
12580:
12576:
12573:
12570:
12564:
12561:
12558:
12554:
12549:
12544:
12538:
12535:
12531:
12527:
12522:
12518:
12511:
12506:
12501:
12496:
12492:
12488:
12485:
12482:
12479:
12473:
12470:
12443:
12440:
12437:
12432:
12426:
12421:
12417:
12411:
12408:
12404:
12399:
12394:
12389:
12386:
12382:
12378:
12375:
12372:
12369:
12366:
12338:
12335:
12332:
12328:
12325:
12319:
12315:
12311:
12308:
12305:
12301:
12296:
12291:
12285:
12281:
12277:
12272:
12269:
12265:
12261:
12256:
12252:
12245:
12240:
12235:
12230:
12226:
12220:
12215:
12212:
12208:
12204:
12201:
12198:
12195:
12189:
12186:
12161:
12158:
12155:
12151:
12148:
12145:
12141:
12138:
12134:
12131:
12128:
12125:
12120:
12116:
12112:
12109:
12106:
12103:
12097:
12094:
12071:
12068:
12065:
12061:
12058:
12055:
12052:
12048:
12045:
12039:
12035:
12031:
12028:
12025:
12022:
12017:
12013:
12000:
11996:
11993:
11981:
11978:
11975:
11970:
11965:
11961:
11958:
11952:
11948:
11945:
11940:
11936:
11933:
11926:
11923:
11920:
11914:
11911:
11905:
11902:
11899:
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11890:
11884:
11879:
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11865:
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11814:
11811:
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11784:
11781:
11778:
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11768:
11765:
11760:
11756:
11752:
11728:
11723:
11719:
11715:
11712:
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11700:
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11670:
11667:
11664:
11661:
11658:
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11650:
11647:
11643:
11637:
11633:
11629:
11626:
11623:
11620:
11617:
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11584:
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11528:
11523:
11519:
11513:
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11483:
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11438:
11435:
11432:
11425:
11421:
11416:
11412:
11409:
11406:
11402:
11399:
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11390:
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11380:
11376:
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11350:
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11331:
11328:
11325:
11322:
11319:
11316:
11313:
11310:
11307:
11304:
11301:
11298:
11293:
11289:
11264:
11261:
11257:
11252:
11248:
11240:
11236:
11231:
11227:
11224:
11221:
11217:
11210:
11206:
11201:
11178:
11175:
11172:
11168:
11164:
11161:
11158:
11155:
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11149:
11146:
11143:
11140:
11135:
11131:
11086:
11069:
11066:
11063:
11059:
11056:
11051:
11047:
11043:
11039:
11033:
11029:
11025:
11020:
11016:
11012:
11009:
11006:
10999:
10995:
10990:
10986:
10983:
10980:
10976:
10969:
10965:
10960:
10935:
10932:
10926:
10922:
10918:
10915:
10908:
10904:
10899:
10895:
10892:
10889:
10863:
10851:The extension
10842:
10827:
10810:
10807:
10802:
10798:
10792:
10788:
10784:
10781:
10778:
10773:
10769:
10763:
10759:
10742:
10723:
10720:
10715:
10711:
10707:
10704:
10701:
10698:
10695:
10692:
10687:
10683:
10679:
10674:
10670:
10666:
10641:Thus, setting
10628:
10623:
10619:
10615:
10612:
10609:
10606:
10603:
10600:
10597:
10594:
10589:
10584:
10580:
10576:
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10567:
10563:
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10557:
10554:
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10548:
10545:
10542:
10539:
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10531:
10526:
10522:
10499:
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10491:
10487:
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10480:
10477:
10474:
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10468:
10465:
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10456:
10453:
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10443:
10439:
10435:
10432:
10427:
10423:
10419:
10416:
10413:
10408:
10404:
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10394:
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10383:
10378:
10374:
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10364:
10359:
10355:
10351:
10346:
10342:
10338:
10335:
10294:
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10288:
10283:
10279:
10275:
10270:
10266:
10262:
10259:
10256:
10253:
10250:
10245:
10241:
10215:
10212:
10209:
10206:
10203:
10200:
10197:
10194:
10191:
10188:
10185:
10182:
10179:
10176:
10173:
10170:
10167:
10156:is defined by
10123:
10120:
10117:
10113:
10110:
10107:
10104:
10101:
10098:
10095:
10090:
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10082:
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10068:
10065:
10062:
10058:
10051:
10047:
10042:
10035:
10032:
10029:
10025:
10022:
10017:
10013:
10009:
10006:
10003:
10000:
9995:
9990:
9987:
9983:
9979:
9976:
9973:
9970:
9967:
9963:
9956:
9952:
9947:
9901:
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9880:
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9869:
9864:
9860:
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9848:
9844:
9837:
9832:
9827:
9823:
9819:
9814:
9809:
9805:
9801:
9796:
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9787:
9783:
9778:
9774:
9747:
9743:
9739:
9736:
9731:
9727:
9723:
9720:
9715:
9711:
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9699:
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9690:
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9672:
9668:
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9633:
9628:
9624:
9620:
9617:
9614:
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9600:
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9591:
9587:
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9574:
9571:
9566:
9562:
9532:
9529:
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9520:
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9467:
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9460:
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9436:
9430:
9426:
9422:
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9416:
9413:
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8827:
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8436:
8433:
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6935:
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6928:
6924:
6921:
6918:
6915:
6912:
6909:
6903:
6900:
6886:is defined by
6873:
6867:
6842:
6837:
6833:
6827:
6823:
6819:
6814:
6810:
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6779:
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6428:
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6240:
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6231:
6227:
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6218:
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6181:
6176:
6169:
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6142:
6137:
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6122:
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6079:
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6065:
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5320:
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5287:
5284:
5279:
5278:
5275:
5272:
5265:
5256:the classical
5239:
5236:
5233:
5229:
5225:
5222:
5219:
5215:
5211:
5208:
5205:
5202:
5197:
5192:
5189:
5185:
5181:
5178:
5175:
5171:
5168:
5146:
5141:
5137:
5133:
5130:
5127:
5124:
5119:
5114:
5110:
5106:
5103:
5100:
5055:
5049:
5024:
5021:
5016:
5011:
5007:
5003:
4998:
4993:
4989:
4985:
4980:
4976:
4972:
4969:
4966:
4939:
4936:
4932:
4929:
4923:
4920:
4916:
4912:
4909:
4906:
4881:
4875:
4871:
4867:
4864:
4859:
4855:
4851:
4847:
4841:
4838:
4834:
4830:
4825:
4821:
4817:
4778:
4772:
4739:
4734:
4731:
4727:
4723:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4699:
4696:
4693:
4690:
4687:
4684:
4681:
4678:
4652:
4646:
4643:
4641:
4638:
4637:
4634:
4631:
4629:
4626:
4625:
4623:
4618:
4615:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4552:
4546:
4521:
4518:
4501:
4496:
4490:
4484:
4479:
4475:
4452:
4449:
4445:
4442:
4439:
4434:
4430:
4426:
4423:
4420:
4415:
4411:
4407:
4404:
4401:
4398:
4393:
4389:
4385:
4381:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4319:
4314:
4308:
4302:
4297:
4293:
4276:
4262:
4259:
4254:
4250:
4246:
4241:
4237:
4233:
4230:
4227:
4222:
4218:
4214:
4211:
4208:
4205:
4202:
4197:
4193:
4166:
4163:
4158:
4152:
4147:
4142:
4138:
4134:
4131:
4127:
4120:
4117:
4113:
4109:
4106:
4103:
4100:
4095:
4091:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4040:
4037:
4034:
4031:
4028:
4023:
4018:
4015:
4012:
4009:
4005:
4001:
3998:
3995:
3992:
3989:
3986:
3981:
3978:
3974:
3970:
3965:
3961:
3936:
3930:
3907:
3889:
3883:
3879:
3876:
3873:
3869:
3865:
3861:
3857:
3854:
3850:
3846:
3843:
3840:
3837:
3833:
3788:
3784:
3780:
3776:
3773:
3769:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3736:
3732:
3728:
3725:
3722:
3719:
3714:
3710:
3704:
3700:
3696:
3693:
3690:
3686:
3680:
3676:
3671:
3667:
3663:
3659:
3654:
3617:
3612:
3608:
3604:
3601:
3598:
3593:
3589:
3585:
3582:
3577:
3573:
3569:
3551:
3533:
3530:
3507:
3504:
3500:
3496:
3493:
3490:
3487:
3482:
3478:
3474:
3471:
3447:
3439:
3436:
3430:
3427:
3424:
3421:
3418:
3417:
3414:
3411:
3408:
3402:
3398:
3394:
3390:
3386:
3381:
3377:
3373:
3370:
3367:
3363:
3359:
3356:
3353:
3350:
3345:
3341:
3337:
3334:
3331:
3330:
3328:
3298:
3295:
3292:
3286:
3282:
3275:
3271:
3268:
3265:
3260:
3253:
3250:
3240:
3237:
3234:
3229:
3222:
3219:
3212:
3209:
3206:
3203:
3197:
3192:
3188:
3180:
3176:
3170:
3166:
3134:
3130:
3125:
3122:
3118:
3114:
3111:
3105:
3102:
3099:
3096:
3091:
3087:
3066:
3060:
3043:
3039:
3035:
3030:
3025:
3021:
3017:
3014:
2992:
2989:
2986:
2980:
2976:
2972:
2969:
2966:
2961:
2957:
2953:
2950:
2947:
2941:
2938:
2932:
2929:
2926:
2923:
2920:
2917:
2893:
2890:
2884:
2881:
2877:
2873:
2870:
2867:
2864:
2861:
2858:
2852:
2849:
2822:
2816:
2813:
2810:
2807:
2802:
2799:
2796:
2793:
2787:
2784:
2781:
2778:
2773:
2769:
2728:
2725:
2717:
2713:
2709:
2703:
2699:
2693:
2690:
2687:
2684:
2679:
2675:
2626:
2623:
2620:
2616:
2613:
2608:
2604:
2599:
2596:
2593:
2590:
2585:
2580:
2577:
2573:
2569:
2566:
2563:
2559:
2556:
2534:
2529:
2525:
2521:
2518:
2515:
2512:
2509:
2504:
2499:
2495:
2491:
2488:
2485:
2440:
2434:
2409:
2404:
2400:
2396:
2393:
2390:
2385:
2380:
2376:
2372:
2367:
2362:
2358:
2354:
2349:
2344:
2340:
2336:
2331:
2327:
2323:
2320:
2317:
2290:
2287:
2283:
2280:
2276:
2273:
2267:
2264:
2260:
2256:
2253:
2250:
2225:
2219:
2215:
2211:
2208:
2203:
2199:
2195:
2192:
2187:
2183:
2179:
2175:
2169:
2166:
2162:
2158:
2153:
2149:
2145:
2116:
2113:
2109:
2105:
2102:
2097:
2091:
2058:
2053:
2050:
2046:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1975:
1969:
1966:
1964:
1961:
1960:
1957:
1954:
1952:
1949:
1948:
1946:
1941:
1938:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1870:
1864:
1837:
1834:
1825:
1821:
1818:Kostant (1969)
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1759:
1755:
1733:The functions
1720:
1717:
1714:
1711:
1708:
1702:
1697:
1692:
1689:
1686:
1683:
1679:
1673:
1669:
1665:
1660:
1656:
1652:
1649:
1629:
1626:
1623:
1618:
1615:
1611:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1523:
1519:
1515:
1511:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1440:
1412:
1402:
1399:
1398:
1397:
1377:
1376:
1344:
1341:
1338:
1335:
1330:
1326:
1321:
1317:
1314:
1311:
1306:
1302:
1298:
1295:
1268:
1264:
1260:
1256:
1253:
1239:
1211:
1208:
1205:
1202:
1197:
1193:
1189:
1185:
1181:
1178:
1173:
1170:
1166:
1162:
1159:
1156:
1153:
1150:
1146:
1143:
1112:
1078:
1067:
1066:
1063:
1062:
1051:
1048:
1045:
1039:
1036:
1032:
1028:
1025:
1022:
1017:
1013:
1007:
1003:
999:
996:
993:
990:
985:
981:
953:
928:
898:
879:
876:
873:
869:
866:
863:
858:
854:
850:
847:
844:
841:
838:
833:
829:
825:
822:
819:
816:
813:
810:
807:
802:
798:
781:
756:
742:
711:
693:is called the
669:
647:
620:
601:
584:
548:
542:
498:
466:
431:
359:
323:Main article:
320:
317:
305:Selberg (1956)
252:is complex or
237:
206:
157:Lorentz groups
134:
131:
34:Harish-Chandra
9:
6:
4:
3:
2:
27375:
27364:
27361:
27359:
27356:
27354:
27351:
27350:
27348:
27333:
27325:
27324:
27321:
27315:
27312:
27310:
27307:
27305:
27304:Weak topology
27302:
27300:
27297:
27295:
27292:
27290:
27287:
27286:
27284:
27280:
27273:
27269:
27266:
27264:
27261:
27259:
27256:
27254:
27251:
27249:
27246:
27244:
27241:
27239:
27236:
27234:
27231:
27229:
27228:Index theorem
27226:
27224:
27221:
27219:
27216:
27214:
27211:
27210:
27208:
27204:
27198:
27195:
27193:
27190:
27189:
27187:
27185:Open problems
27183:
27177:
27174:
27172:
27169:
27167:
27164:
27162:
27159:
27157:
27154:
27152:
27149:
27148:
27146:
27142:
27136:
27133:
27131:
27128:
27126:
27123:
27121:
27118:
27116:
27113:
27111:
27108:
27106:
27103:
27101:
27098:
27096:
27093:
27091:
27088:
27087:
27085:
27081:
27075:
27072:
27070:
27067:
27065:
27062:
27060:
27057:
27055:
27052:
27050:
27047:
27045:
27042:
27040:
27037:
27035:
27032:
27031:
27029:
27027:
27023:
27013:
27010:
27008:
27005:
27003:
27000:
26997:
26993:
26989:
26986:
26984:
26981:
26979:
26976:
26975:
26973:
26969:
26963:
26960:
26958:
26955:
26953:
26950:
26948:
26945:
26943:
26940:
26938:
26935:
26933:
26930:
26928:
26925:
26923:
26920:
26918:
26915:
26914:
26911:
26908:
26904:
26899:
26895:
26891:
26884:
26879:
26877:
26872:
26870:
26865:
26864:
26861:
26853:
26848:
26844:
26841:(in French),
26840:
26835:
26831:
26826:
26822:
26818:
26817:
26811:
26807:
26803:
26799:
26798:Selberg, Atle
26795:
26792:
26788:
26783:
26778:
26774:
26770:
26769:
26764:
26760:
26756:
26751:
26748:
26746:0-691-08184-0
26742:
26738:
26734:
26733:Lax, Peter D.
26730:
26727:
26725:0-387-96198-4
26721:
26717:
26713:
26709:
26705:
26700:
26696:
26692:
26685:
26681:
26677:
26674:
26672:0-691-09089-0
26668:
26664:
26660:
26656:
26653:
26651:0-387-95115-6
26647:
26643:
26639:
26634:
26631:
26627:
26623:
26619:
26615:
26610:
26609:
26604:
26602:0-12-338460-5
26598:
26594:
26589:
26586:
26584:0-12-338301-3
26580:
26575:
26574:
26567:
26562:
26557:
26553:
26548:
26544:
26540:
26539:
26534:
26530:
26528:, section 21.
26526:
26522:
26517:
26512:
26508:
26504:
26500:
26495:
26492:
26488:
26484:
26480:
26476:
26472:
26471:
26465:
26462:
26458:
26454:
26450:
26446:
26442:
26438:
26434:
26433:
26427:
26424:
26420:
26415:
26410:
26406:
26402:
26397:
26392:
26388:
26384:
26379:
26376:
26372:
26367:
26362:
26358:
26354:
26349:
26344:
26340:
26336:
26332:
26328:
26324:
26320:
26317:
26312:
26310:0-8218-1633-0
26306:
26302:
26298:
26294:
26290:
26286:
26282:
26278:
26274:
26271:
26267:
26263:
26259:
26255:
26251:
26246:
26242:
26238:
26234:
26230:
26222:
26218:
26214:
26210:
26206:
26205:
26200:
26196:
26192:
26189:
26187:0-8218-0711-0
26183:
26179:
26174:
26171:
26167:
26163:
26159:
26154:
26151:
26149:0-12-215506-8
26145:
26141:
26137:
26133:
26130:
26128:0-521-40997-7
26124:
26120:
26116:
26115:Davies, E. B.
26112:
26109:
26105:
26101:
26097:
26093:
26088:
26085:
26081:
26077:
26073:
26069:
26065:
26060:
26056:
26051:
26047:
26043:
26038:
26037:
26027:, p. 489
26026:
26025:Helgason 1984
26021:
26015:, p. 347
26014:
26009:
26002:
26001:Helgason 1984
25997:
25991:
25986:
25979:
25978:Helgason 1984
25974:
25967:
25961:
25955:, p. 437
25954:
25953:Helgason 1984
25949:
25943:, p. 414
25942:
25941:Helgason 1978
25937:
25931:, p. 484
25930:
25929:Helgason 1984
25925:
25919:, p. 407
25918:
25917:Helgason 1978
25913:
25907:, p. 182
25906:
25901:
25895:, p. 177
25894:
25889:
25883:, Chapter VII
25882:
25877:
25871:, p. 447
25870:
25869:Helgason 1984
25865:
25859:, p. 436
25858:
25857:Helgason 1984
25853:
25847:, p. 403
25846:
25845:Helgason 1978
25841:
25835:, p. 435
25834:
25833:Helgason 1984
25829:
25823:, p. 430
25822:
25821:Helgason 1984
25817:
25811:, p. 267
25810:
25809:Helgason 1984
25805:
25799:, p. 447
25798:
25797:Helgason 1984
25793:
25787:, p. 4–5
25786:
25785:Beerends 1987
25781:
25775:
25774:Helgason 1984
25770:
25763:
25759:
25755:
25751:
25744:
25740:
25736:
25732:
25725:
25719:
25713:
25709:
25702:
25698:
25693:
25686:
25685:Helgason 1984
25681:
25675:, p. 133
25674:
25669:
25663:, p. 133
25662:
25657:
25650:
25632:
25628:
25624:
25621:
25613:
25607:
25602:
25598:
25589:
25583:
25577:
25576:Helgason 1978
25572:
25566:, p. 115
25565:
25560:
25553:
25552:Helgason 1984
25548:
25539:
25533:
25528:
25522:
25517:
25510:
25509:Helgason 1984
25505:
25498:
25497:Helgason 1984
25493:
25487:
25482:
25476:
25471:
25465:
25460:
25458:
25456:
25448:
25447:Helgason 1984
25443:
25437:
25432:
25426:
25421:
25414:
25410:
25409:measure class
25404:
25373:
25369:
25363:
25356:
25351:
25345:
25340:
25334:
25329:
25327:
25320:
25315:
25313:
25311:
25304:
25299:
25293:
25288:
25282:
25277:
25271:
25266:
25259:
25258:Helgason 1984
25254:
25250:
25243:
25241:
25236:
25231:
25218:
25213:
25196:
25192:
25189:the space of
25176:
25171:
25161:
25133:
25120:
25116:
25109:
25105:
25098:
25092:
25087:
25079:
25076:
25065:
25054:
25051:
25048:
25042:
25039:
25036:
25029:
25023:
25015:
25010:
25002:
24998:
24994:
24982:
24978:
24975:
24971:
24967:
24961:
24957:
24953:
24947:
24929:
24921:
24895:
24891:
24886:
24873:
24870:
24867:
24861:
24858:
24845:
24839:
24829:
24825:
24818:
24806:
24796:
24791:
24778:
24774:
24771:
24768:
24748:
24745:
24742:
24735:
24732:
24727:
24724:
24720:
24713:
24708:
24704:
24700:
24694:
24686:
24682:
24672:
24659:
24653:
24647:
24644:
24641:
24635:
24629:
24621:
24617:
24613:
24609:
24593:
24590:
24584:
24581:
24568:
24562:
24551:
24539:
24529:
24524:
24511:
24507:
24501:
24495:
24486:
24484:
24480:
24476:
24472:
24464:
24446:
24441:
24412:
24409:
24406:
24403:
24398:
24367:
24350:
24346:
24342:
24337:
24335:
24326:
24324:
24319:
24317:
24312:
24310:
24306:
24302:
24298:
24295:In this case
24293:
24280:
24274:
24270:
24267:
24264:
24260:
24256:
24252:
24246:
24243:
24230:
24222:
24219:
24211:
24207:
24203:
24192:
24180:
24164:
24162:
24158:
24154:
24150:
24146:
24142:
24132:
24129:
24125:
24121:
24117:
24112:
24099:
24095:
24088:
24085:
24080:
24077:
24073:
24069:
24064:
24060:
24050:
24047:
24040:
24028:
24025:
24021:
24017:
24014:
24010:
24004:
24000:
23995:
23991:
23986:
23982:
23961:
23954:
23949:
23940:
23936:
23932:
23929:
23923:
23920:
23915:
23912:
23908:
23904:
23899:
23895:
23889:
23886:
23880:
23874:
23871:
23865:
23856:
23851:
23842:
23838:
23834:
23831:
23825:
23822:
23819:
23816:
23811:
23807:
23801:
23798:
23792:
23786:
23783:
23777:
23759:
23755:
23751:
23748:
23742:
23726:
23722:
23718:
23715:
23709:
23706:
23702:
23693:
23689:
23685:
23679:
23669:
23665:
23660:
23647:
23643:
23640:
23636:
23632:
23624:
23623:
23622:
23620:
23616:
23613:-function of
23612:
23607:
23603:
23599:
23595:
23573:
23570:
23553:
23549:
23542:
23538:
23533:
23529:
23524:
23511:
23503:
23499:
23492:
23489:
23481:
23477:
23470:
23467:
23459:
23455:
23449:
23445:
23438:
23415:
23405:
23401:
23396:
23389:
23384:
23380:
23369:
23365:
23360:
23356:
23350:
23340:
23336:
23330:
23326:
23321:
23312:
23308:
23304:
23300:
23280:
23275:
23271:
23264:
23256:
23252:
23248:
23243:
23239:
23232:
23229:
23226:
23220:
23212:
23207:
23194:
23188:
23178:
23174:
23169:
23165:
23159:
23153:
23145:
23143:
23125:
23120:
23108:
23086:
23081:
23061:
23057:
23053:
23049:
23045:
23041:
23036:
23018:
23014:
23007:
23004:
22996:
22992:
22985:
22982:
22974:
22970:
22964:
22960:
22953:
22933:
22927:
22924:
22919:
22915:
22908:
22902:
22897:
22893:
22889:
22884:
22880:
22873:
22870:
22864:
22861:
22856:
22852:
22846:
22842:
22835:
22826:
22813:
22807:
22804:
22801:
22795:
22789:
22781:
22778:
22774:
22770:
22764:
22756:
22752:
22745:
22742:
22739:
22733:
22713:
22710:
22707:
22700:
22697:
22694:
22688:
22683:
22680:
22675:
22672:
22668:
22664:
22658:
22652:
22648:
22644:
22638:
22632:
22626:
22623:
22620:
22614:
22606:
22602:
22598:
22594:
22590:
22586:
22582:
22578:
22560:
22543:
22539:
22535:
22531:
22527:
22523:
22519:
22515:
22510:
22506:
22502:
22494:
22490:
22482:
22477:
22473:
22469:
22465:
22459:
22450:
22449:Bruhat (1956)
22446:
22442:
22439:-function in
22438:
22432:
22422:
22409:
22406:
22403:
22390:
22383:
22380:
22368:
22362:
22358:
22354:
22348:
22342:
22316:
22299:
22294:
22281:
22278:
22273:
22270:
22267:
22263:
22259:
22254:
22251:
22247:
22235:
22221:
22195:
22179:
22176:
22173:
22161:
22158:
22154:
22150:
22145:
22141:
22133:
22130:
22116:
22109:
22106:
22092:
22086:
22082:
22076:
22073:
22070:
22067:
22063:
22059:
22051:
22047:
22038:
22034:
22025:
22020:
22007:
22004:
22001:
21995:
21992:
21984:
21981:
21974:
21971:
21965:
21961:
21957:
21953:
21949:
21943:
21935:
21931:
21922:
21921:
21917:
21909:
21905:
21901:
21897:
21893:
21875:
21863:
21841:
21824:
21817:
21814:
21810:
21806:
21801:
21788:
21785:
21782:
21779:
21774:
21771:
21768:
21764:
21760:
21757:
21749:
21745:
21741:
21732:
21730:
21712:
21695:
21677:
21674:
21668:
21665:
21662:
21659:
21652:
21645:
21639:
21636:
21630:
21625:
21621:
21612:
21608:
21585:
21564:
21560:
21555:
21542:
21537:
21534:
21530:
21523:
21520:
21514:
21509:
21506:
21503:
21499:
21495:
21490:
21486:
21477:
21470:
21461:
21435:
21431:
21427:
21423:
21416:
21412:
21408:
21390:
21370:
21365:
21352:
21347:
21344:
21341:
21338:
21334:
21328:
21325:
21322:
21318:
21314:
21311:
21308:
21305:
21302:
21299:
21296:
21288:
21281:
21265:
21260:
21257:
21253:
21246:
21238:
21234:
21225:
21222:
21218:
21212:
21209:
21206:
21203:
21199:
21195:
21190:
21186:
21170:
21166:
21162:
21157:
21131:
21127:
21109:
21105:
21100:
21094:
21086:
21078:
21073:
21065:
21056:
21052:
21047:
21043:
21026:
21021:
21017:
21001:
20996:
20992:
20986:
20983:
20980:
20976:
20972:
20957:
20952:
20948:
20927:
20922:
20918:
20911:
20908:
20905:
20902:
20899:
20890:
20886:
20880:
20877:
20874:
20870:
20866:
20861:
20857:
20853:
20850:
20841:
20838:
20834:
20792:
20787:
20783:
20779:
20776:
20773:
20738:
20732:
20729:
20723:
20720:
20715:
20711:
20672:
20656:
20651:
20647:
20642:
20639:
20636:
20630:
20626:
20620:
20617:
20614:
20610:
20606:
20591:
20588:
20580:
20576:
20563:
20558:
20555:
20552:
20547:
20540:
20537:
20534:
20531:
20525:
20519:
20516:
20512:
20502:
20496:
20490:
20487:
20455:
20451:
20448:, called the
20447:
20419:
20415:
20411:
20407:
20403:
20375:
20371:
20367:
20363:
20359:
20355:
20345:
20325:
20299:
20295:
20291:
20285:
20277:
20273:
20269:
20266:
20243:
20240:
20234:
20231:
20218:
20212:
20204:
20199:
20194:
20177:
20173:
20163:
20159:
20155:
20149:
20141:
20137:
20127:
20118:
20109:
20106:
20103:
20095:
20090:
20077:
20074:
20069:
20065:
20057:
20049:
20046:
20042:
20036:
20033:
20028:
20025:
20017:
20013:
20006:
19990:
19985:
19982:
19974:
19971:
19968:
19962:
19957:
19953:
19928:
19923:
19914:
19910:
19906:
19903:
19897:
19894:
19889:
19886:
19882:
19878:
19873:
19869:
19863:
19860:
19854:
19848:
19845:
19839:
19830:
19825:
19816:
19812:
19808:
19805:
19799:
19796:
19793:
19790:
19785:
19781:
19775:
19772:
19766:
19760:
19757:
19751:
19733:
19729:
19725:
19722:
19716:
19700:
19696:
19692:
19689:
19683:
19680:
19676:
19665:
19660:
19652:
19649:
19645:
19641:
19636:
19632:
19628:
19622:
19616:
19608:
19605:
19589:
19586:
19583:
19577:
19574:
19561:
19555:
19543:
19535:
19531:
19524:
19512:
19502:
19497:
19484:
19480:
19474:
19468:
19460:
19455:
19442:
19439:
19436:
19429:
19421:
19418:
19414:
19407:
19401:
19396:
19392:
19388:
19382:
19370:
19359:
19355:
19351:
19343:
19325:
19303:
19297:
19279:
19274:
19242:
19234:
19230:
19225:
19218:
19210:
19206:
19202:
19199:
19191:
19187:
19145:
19080:
19076:
19072:
19044:
19040:
19022:
19010:
19007:
19004:
18999:
18995:
18986:
18967:
18961:
18956:
18940:If α ≠ 0 and
18938:
18925:
18909:
18906:
18898:
18892:
18886:
18883:
18877:
18874:
18871:
18865:
18855:
18852:
18846:
18841:
18824:
18808:
18792:and for α in
18742:
18729:
18726:
18723:
18713:
18710:
18707:
18704:
18699:
18689:
18679:
18664:
18648:
18636:
18621:
18603:
18591:
18538:
18521:
18493:
18483:
18481:
18477:
18449:
18444:
18431:
18422:
18419:
18416:
18405:
18402:
18399:
18393:
18387:
18384:
18379:
18376:
18373:
18369:
18365:
18360:
18357:
18349:
18346:
18340:
18337:
18334:
18331:
18325:
18319:
18311:
18307:
18303:
18299:
18295:
18291:
18286:
18273:
18270:
18267:
18261:
18248:
18245:
18239:
18229:
18211:
18204:
18198:
18190:
18182:
18178:
18171:
18159:
18149:
18144:
18131:
18127:
18124:
18121:
18115:
18102:
18099:
18093:
18083:
18065:
18058:
18050:
18047:
18037:
18033:
18026:
18023:
18011:
18001:
17996:
17983:
17979:
17973:
17967:
17962:
17958:
17954:
17948:
17942:
17922:
17919:
17916:
17910:
17897:
17894:
17888:
17880:
17869:
17863:
17855:
17850:
17832:
17823:
17815:
17811:
17804:
17792:
17782:
17777:
17764:
17760:
17754:
17748:
17737:
17733:
17729:
17725:
17721:
17717:
17713:
17708:
17695:
17689:
17686:
17674:
17668:
17662:
17654:
17646:
17641:
17637:
17624:
17611:
17608:
17605:
17598:
17590:
17587:
17577:
17573:
17569:
17563:
17555:
17552:
17542:
17538:
17534:
17528:
17522:
17511:
17489:
17486:
17476:
17472:
17460:
17456:
17451:
17438:
17433:
17429:
17422:
17419:
17414:
17404:
17400:
17390:
17377:
17371:
17366:
17362:
17355:
17352:
17345:
17341:
17336:
17330:
17326:
17321:
17317:
17314:
17311:
17305:
17302:
17294:
17290:
17286:
17282:
17273:
17269:
17264:
17251:
17248:
17245:
17238:
17235:
17229:
17224:
17220:
17216:
17208:
17204:
17197:
17192:
17188:
17162:
17158:
17153:
17147:
17143:
17134:
17130:
17116:
17112:
17103:
17099:
17080:
17076:
17072:
17066:
17053:
17049:
17040:
17035:
17022:
17019:
17014:
17006:
17000:
16997:
16989:
16985:
16974:
16967:
16962:
16957:
16953:
16949:
16945:
16941:
16937:
16932:
16919:
16916:
16913:
16904:
16900:
16896:
16890:
16885:
16881:
16877:
16871:
16865:
16862:
16839:
16835:
16831:
16825:
16812:
16808:
16799:
16778:
16774:
16769:
16763:
16759:
16750:
16746:
16732:
16728:
16719:
16715:
16711:
16707:
16700:
16693:
16686:
16679:
16672:
16665:
16658:
16654:
16650:
16646:
16641:
16626:
16622:
16618:
16613:
16609:
16604:
16598:
16594:
16585:
16581:
16571:
16558:
16553:
16549:
16545:
16542:
16538:
16534:
16528:
16493:
16477:
16474:
16469:
16465:
16461:
16458:
16455:
16447:
16440:
16436:
16429:
16422:
16415:
16408:
16401:
16397:
16390:
16383:
16379:
16375:
16371:
16364:
16360:
16356:
16352:
16345:
16335:
16332:
16319:
16314:
16306:
16300:
16291:
16285:
16277:
16269:
16265:
16255:
16243:
16235:
16231:
16227:
16221:
16215:
16207:
16205:
16201:
16196:
16183:
16180:
16170:
16164:
16151:
16143:
16139:
16127:
16121:
16116:
16112:
16108:
16102:
16090:
16079:
16077:
16073:
16069:
16065:
16062:is played by
16061:
16043:
16038:
16021:
16013:
16009:
16005:
16001:
15997:
15993:
15989:
15985:
15982:
15977:
15958:
15954:
15947:
15939:
15935:
15928:
15920:
15901:
15889:
15874:
15868:
15845:
15837:
15831:
15828:
15824:
15817:
15793:
15790:
15787:
15783:
15779:
15771:
15767:
15758:
15745:
15741:
15734:
15728:
15722:
15690:
15686:
15668:
15663:
15645:
15632:
15629:
15625:
15619:
15606:
15600:
15589:
15581:
15570:
15558:
15548:
15543:
15530:
15526:
15523:
15519:
15513:
15500:
15494:
15483:
15475:
15464:
15452:
15442:
15429:
15417:
15408:
15403:
15397:
15391:
15383:
15380:
15367:
15364:
15357:
15349:
15346:
15335:
15329:
15324:
15320:
15316:
15310:
15298:
15287:
15285:
15281:
15276:
15274:
15270:
15266:
15262:
15258:
15253:
15240:
15231:
15225:
15215:
15211:
15202:
15198:
15191:
15183:
15179:
15170:
15157:
15141:
15129:
15126:
15111:
15106:
15085:
15082:
15066:
15053:
15050:
15047:
15041:
15026:
15022:
15015:
15001:
14997:
14990:
14979:
14967:
14958:
14953:
14950:
14947:
14940:
14934:
14929:
14925:
14916:
14914:
14910:
14906:
14901:
14899:
14871:
14855:
14851:
14844:
14836:
14832:
14828:
14825:
14818:
14814:
14809:
14796:
14785:
14781:
14774:
14764:
14761:
14755:
14752:
14748:
14741:
14717:
14714:
14711:
14707:
14700:
14692:
14688:
14679:
14675:
14666:
14650:
14614:
14609:
14591:
14578:
14573:
14569:
14565:
14562:
14559:
14553:
14547:
14543:
14530:
14527:
14521:
14513:
14509:
14500:
14496:
14492:
14489:
14486:
14478:
14474:
14465:
14461:
14452:
14450:
14432:
14415:
14406:
14393:
14390:
14387:
14384:
14381:
14378:
14375:
14372:
14369:
14366:
14358:
14356:
14353:there is the
14351:
14338:
14328:
14325:
14322:
14319:
14316:
14312:
14302:
14299:
14296:
14293:
14290:
14282:
14281:Then setting
14268:
14253:
14249:
14248:maximal torus
14245:
14229:
14219:
14216:
14206:
14143:
14139:
14135:
14125:
14095:
14090:
14088:
14084:
14080:
14076:
14071:
14067:
14063:
14053:
14040:
14037:
14034:
14029:
14024:
14020:
14017:
14011:
14007:
14004:
13999:
13995:
13992:
13983:
13975:
13971:
13964:
13952:
13939:
13935:
13927:
13923:
13919:
13914:
13908:
13902:
13894:
13889:
13885:
13881:
13877:
13872:
13859:
13853:
13845:
13841:
13837:
13831:
13819:
13813:
13807:
13799:
13789:
13777:
13774:
13759:
13755:
13748:
13736:
13730:
13725:
13721:
13714:
13706:
13702:
13693:
13691:
13687:
13683:
13679:
13672:
13654:
13636:
13623:
13620:
13617:
13610:
13607:
13604:
13598:
13593:
13589:
13585:
13579:
13571:
13567:
13558:
13556:
13552:
13548:
13543:
13530:
13527:
13524:
13519:
13514:
13510:
13507:
13501:
13497:
13494:
13489:
13485:
13482:
13473:
13461:
13445:
13441:
13432:
13428:
13424:
13420:
13415:
13412:
13409:
13403:
13400:
13395:
13392:
13386:
13383:
13380:
13375:
13371:
13367:
13364:
13361:
13358:
13342:
13338:
13334:
13331:
13327:
13321:
13309:
13296:
13292:
13283:
13279:
13275:
13271:
13266:
13260:
13254:
13246:
13243:
13230:
13227:
13224:
13217:
13214:
13208:
13205:
13199:
13196:
13184:
13171:
13167:
13161:
13158:
13153:
13147:
13144:
13141:
13135:
13127:
13125:
13103:
13092:
13087:
13074:
13068:
13062:
13059:
13056:
13053:
13050:
13047:
13043:
13039:
13033:
13028:
13024:
13018:
13015:
13011:
13005:
13002:
12991:
12987:
12982:
12979:
12975:
12972:
12969:
12965:
12962:
12957:
12951:
12945:
12941:
12937:
12932:
12928:
12921:
12918:
12914:
12908:
12905:
12891:
12887:
12873:
12869:
12865:
12862:
12859:
12854:
12848:
12843:
12839:
12833:
12830:
12826:
12820:
12817:
12803:
12799:
12790:
12787:
12774:
12771:
12768:
12763:
12757:
12752:
12748:
12742:
12739:
12735:
12729:
12726:
12712:
12708:
12701:
12698:
12693:
12690:
12684:
12678:
12672:
12664:
12662:
12658:
12654:
12649:
12636:
12633:
12630:
12624:
12621:
12618:
12615:
12611:
12604:
12601:
12598:
12592:
12582:
12578:
12574:
12571:
12568:
12562:
12559:
12556:
12552:
12547:
12542:
12536:
12533:
12529:
12525:
12520:
12516:
12509:
12504:
12494:
12490:
12486:
12480:
12468:
12457:
12454:
12441:
12438:
12435:
12430:
12424:
12419:
12415:
12409:
12406:
12402:
12397:
12384:
12380:
12376:
12370:
12364:
12356:
12354:
12349:
12336:
12333:
12330:
12326:
12323:
12317:
12313:
12309:
12306:
12303:
12299:
12294:
12289:
12283:
12279:
12275:
12270:
12267:
12263:
12259:
12254:
12250:
12243:
12238:
12228:
12224:
12210:
12206:
12202:
12196:
12184:
12172:
12159:
12156:
12153:
12146:
12139:
12136:
12129:
12123:
12118:
12114:
12110:
12104:
12092:
12069:
12066:
12063:
12056:
12053:
12046:
12043:
12037:
12033:
12029:
12023:
12015:
12011:
11992:
11979:
11976:
11973:
11968:
11963:
11959:
11956:
11950:
11946:
11943:
11938:
11934:
11931:
11921:
11909:
11900:
11892:
11888:
11874:
11870:
11866:
11860:
11854:
11846:
11842:
11838:
11834:
11830:
11825:
11812:
11806:
11803:
11791:
11785:
11779:
11771:
11763:
11758:
11754:
11739:
11726:
11721:
11717:
11713:
11710:
11705:
11702:
11677:
11674:
11671:
11668:
11665:
11662:
11656:
11648:
11645:
11635:
11631:
11627:
11621:
11615:
11593:
11589:
11582:
11576:
11573:
11568:
11565:
11555:
11551:
11542:
11539:
11526:
11521:
11517:
11511:
11503:
11500:
11495:
11485:
11481:
11471:
11458:
11455:
11452:
11445:
11442:
11439:
11433:
11430:
11423:
11419:
11414:
11410:
11407:
11404:
11400:
11397:
11391:
11388:
11378:
11374:
11369:
11348:
11342:
11337:
11333:
11326:
11323:
11320:
11314:
11311:
11305:
11302:
11291:
11287:
11278:
11262:
11259:
11255:
11250:
11246:
11238:
11234:
11229:
11225:
11222:
11219:
11215:
11208:
11204:
11199:
11189:
11176:
11170:
11166:
11162:
11156:
11153:
11150:
11144:
11138:
11133:
11129:
11120:
11116:
11112:
11108:
11104:
11100:
11096:
11092:
11085:
11080:
11067:
11064:
11061:
11057:
11054:
11049:
11045:
11041:
11031:
11027:
11023:
11018:
11014:
11010:
11007:
10997:
10993:
10988:
10984:
10981:
10978:
10974:
10967:
10963:
10958:
10949:
10933:
10930:
10924:
10920:
10916:
10913:
10906:
10902:
10897:
10893:
10890:
10887:
10879:
10875:
10871:
10866:
10862:
10858:
10854:
10849:
10847:
10841:
10837:
10833:
10826:
10821:
10808:
10805:
10800:
10790:
10786:
10782:
10779:
10776:
10771:
10767:
10761:
10748:
10741:
10737:
10721:
10718:
10713:
10705:
10702:
10699:
10696:
10693:
10685:
10681:
10677:
10672:
10656:
10652:
10648:
10644:
10639:
10626:
10621:
10610:
10607:
10601:
10598:
10595:
10592:
10587:
10582:
10574:
10569:
10558:
10555:
10552:
10549:
10546:
10543:
10540:
10534:
10529:
10524:
10510:
10497:
10494:
10489:
10478:
10475:
10472:
10469:
10466:
10463:
10460:
10454:
10451:
10446:
10441:
10433:
10430:
10425:
10417:
10414:
10411:
10406:
10398:
10395:
10392:
10389:
10386:
10381:
10376:
10368:
10365:
10357:
10353:
10349:
10344:
10336:
10325:
10323:
10319:
10315:
10311:
10306:
10292:
10289:
10281:
10277:
10273:
10268:
10257:
10254:
10251:
10248:
10243:
10230:
10227:
10213:
10207:
10204:
10201:
10195:
10192:
10186:
10183:
10180:
10177:
10174:
10168:
10165:
10157:
10155:
10151:
10147:
10143:
10139:
10134:
10121:
10118:
10115:
10111:
10108:
10105:
10099:
10093:
10080:
10076:
10072:
10069:
10066:
10063:
10060:
10056:
10049:
10045:
10040:
10033:
10030:
10027:
10023:
10020:
10015:
10011:
10004:
9998:
9985:
9981:
9977:
9974:
9971:
9968:
9965:
9961:
9954:
9950:
9945:
9936:
9934:
9930:
9926:
9922:
9918:
9913:
9899:
9894:
9886:
9883:
9878:
9870:
9867:
9862:
9854:
9851:
9846:
9842:
9835:
9830:
9825:
9817:
9812:
9807:
9799:
9794:
9789:
9781:
9776:
9772:
9763:
9760:
9745:
9737:
9734:
9729:
9721:
9718:
9713:
9709:
9702:
9697:
9692:
9684:
9679:
9674:
9666:
9661:
9657:
9648:
9645:
9631:
9626:
9622:
9615:
9612:
9609:
9603:
9598:
9593:
9589:
9585:
9580:
9576:
9572:
9569:
9564:
9551:
9549:
9544:
9530:
9527:
9524:
9521:
9518:
9510:
9506:
9502:
9498:
9494:
9490:
9486:
9481:
9468:
9465:
9462:
9458:
9455:
9449:
9445:
9441:
9438:
9428:
9424:
9420:
9417:
9411:
9408:
9405:
9399:
9396:
9393:
9389:
9386:
9382:
9379:
9373:
9369:
9365:
9362:
9352:
9348:
9344:
9341:
9335:
9332:
9329:
9321:
9319:
9315:
9311:
9307:
9303:
9299:
9295:
9291:
9287:
9283:
9279:
9275:
9271:
9267:
9263:
9259:
9255:
9250:
9246:
9228:
9224:
9220:
9215:
9211:
9207:
9202:
9198:
9194:
9191:
9188:
9183:
9179:
9171:
9166:
9153:
9148:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9118:
9115:
9112:
9109:
9106:
9098:
9095:
9092:
9089:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9062:
9059:
9056:
9046:
9043:
9040:
9037:
9034:
9031:
9028:
9025:
9022:
9019:
9016:
9010:
9007:
9004:
8999:
8996:
8993:
8990:
8987:
8984:
8981:
8978:
8975:
8972:
8969:
8966:
8963:
8957:
8952:
8949:
8946:
8941:
8937:
8927:
8914:
8909:
8905:
8901:
8898:
8895:
8892:
8889:
8886:
8864:
8858:
8855:
8852:
8847:
8844:
8841:
8838:
8831:
8828:
8825:
8822:
8817:
8814:
8811:
8805:
8800:
8797:
8789:
8785:
8781:
8776:
8763:
8758:
8752:
8749:
8746:
8741:
8738:
8735:
8732:
8729:
8722:
8719:
8716:
8713:
8708:
8705:
8702:
8696:
8691:
8686:
8682:
8673:
8668:
8664:
8646:
8630:parameter in
8629:
8619:
8617:
8596:
8579:
8575:
8565:
8552:
8549:
8542:
8534:
8530:
8523:
8515:
8511:
8504:
8498:
8493:
8489:
8485:
8481:
8477:
8471:
8468:
8465:
8459:
8456:
8448:
8434:
8431:
8426:
8421:
8417:
8414:
8408:
8404:
8401:
8396:
8392:
8389:
8379:with measure
8358:
8349:
8346:
8335:
8331:
8319:
8302:
8298:
8294:
8291:
8278:
8265:
8257:
8253:
8249:
8244:
8240:
8233:
8225:
8221:
8214:
8208:
8200:
8187:
8184:
8169:
8166:
8161:
8155:
8145:
8137:
8134:
8130:
8123:
8120:
8116:
8112:
8106:
8098:
8094:
8085:
8084:The function
8082:
8069:
8060:
8054:
8048:
8043:
8040:
8037:
8034:
8031:
8021:
8018:
8015:
8012:
8004:
7998:
7992:
7984:
7981:
7977:
7968:
7964:
7955:
7939:
7922:
7918:
7910:
7905:
7885:
7879:
7874:
7866:
7861:
7856:
7852:
7840:
7835:
7822:
7812:
7801:
7797:
7793:
7790:
7779:
7773:
7760:
7758:
7754:
7750:
7746:
7741:
7728:
7720:
7713:
7709:
7702:
7694:
7690:
7683:
7677:
7672:
7661:
7658:
7650:
7640:
7636:
7632:
7629:
7622:
7616:
7607:
7603:
7599:
7595:
7580:
7578:
7574:
7570:
7566:
7553:
7548:
7544:
7539:
7532:
7529:
7523:
7518:
7514:
7509:
7505:
7500:
7496:
7490:
7477:
7475:
7474:eigenfunction
7466:
7453:
7450:
7447:
7442:
7437:
7433:
7430:
7424:
7420:
7417:
7412:
7408:
7405:
7387:
7383:
7373:
7363:
7346:
7342:
7338:
7335:
7332:
7321:
7317:
7309:
7305:
7297:
7284:
7275:
7273:
7262:
7258:
7253:
7237:
7219:
7206:
7203:
7200:
7195:
7190:
7186:
7183:
7177:
7173:
7170:
7165:
7161:
7158:
7149:
7137:
7128:
7120:
7116:
7102:
7098:
7094:
7088:
7082:
7074:
7072:
7068:
7064:
7057:
7053:
7048:
7035:
7029:
7017:
7011:
7005:
6997:
6989:
6984:
6979:
6975:
6961:
6958:
6945:
6942:
6939:
6933:
6930:
6926:
6922:
6919:
6916:
6910:
6898:
6887:
6871:
6853:
6840:
6835:
6831:
6825:
6821:
6817:
6812:
6799:
6796:
6783:
6780:
6777:
6771:
6768:
6765:
6762:
6759:
6754:
6750:
6747:
6744:
6741:
6738:
6735:
6732:
6729:
6726:
6723:
6719:
6712:
6709:
6704:
6700:
6693:
6690:
6686:
6681:
6675:
6670:
6666:
6657:
6653:
6644:
6642:
6638:
6634:
6630:
6621:
6606:
6596:
6592:
6586:
6583:
6575:
6569:
6566:
6561:
6557:
6548:
6546:
6542:
6538:
6522:
6519:
6516:
6512:
6506:
6503:
6500:
6496:
6486:
6473:
6468:
6464:
6461:
6455:
6452:
6447:
6444:
6439:
6436:
6433:
6426:
6423:
6420:
6417:
6412:
6409:
6406:
6403:
6397:
6394:
6388:
6380:
6376:
6372:
6366:
6360:
6351:
6338:
6333:
6329:
6324:
6317:
6314:
6308:
6303:
6299:
6294:
6290:
6285:
6281:
6275:
6248:
6238:
6233:
6229:
6225:
6220:
6216:
6208:The function
6206:
6193:
6188:
6183:
6179:
6174:
6167:
6164:
6158:
6153:
6144:
6140:
6135:
6125:
6120:
6116:
6107:
6103:
6087:
6084:
6077:
6072:
6067:
6063:
6056:
6053:
6046:
6033:
6029:
6026:
6023:
6019:
6016:
6010:
6005:
6001:
5990:
5977:
5968:
5964:
5957:
5952:
5939:
5936:
5933:
5926:
5921:
5916:
5912:
5905:
5902:
5895:
5882:
5878:
5875:
5872:
5868:
5865:
5859:
5854:
5850:
5839:
5826:
5817:
5814:
5801:
5798:
5791:
5788:
5785:
5782:
5779:
5773:
5760:
5756:
5750:
5746:
5742:
5738:
5734:
5728:
5725:
5722:
5716:
5711:
5706:
5702:
5693:
5692:* defined by
5688:
5683:
5670:
5666:
5659:
5656:
5650:
5645:
5636:
5630:
5626:
5622:
5617:
5613:
5607:
5594:
5592:
5586:
5579:
5571:
5566:
5550:
5532:
5526:
5513:
5510:
5506:
5499:
5496:
5490:
5485:
5476:
5470:
5466:
5462:
5459:
5454:
5450:
5444:
5431:
5428:
5412:
5409:
5406:
5400:
5397:
5392:
5388:
5384:
5380:
5376:
5370:
5367:
5364:
5361:
5358:
5352:
5347:
5343:
5334:
5318:
5301:
5297:
5293:
5283:
5276:
5273:
5270:
5266:
5263:
5259:
5255:
5254:
5253:
5250:
5237:
5234:
5231:
5227:
5223:
5220:
5217:
5213:
5206:
5200:
5187:
5183:
5179:
5176:
5173:
5169:
5166:
5157:
5144:
5139:
5131:
5128:
5125:
5122:
5117:
5112:
5104:
5101:
5090:
5086:
5083:in SO(2) and
5082:
5078:
5075:
5071:
5053:
5035:
5022:
5014:
5009:
5001:
4996:
4991:
4978:
4974:
4970:
4967:
4956:
4955:
4950:
4937:
4934:
4930:
4927:
4921:
4918:
4914:
4910:
4907:
4904:
4896:
4893:
4879:
4873:
4869:
4865:
4862:
4857:
4853:
4849:
4845:
4839:
4836:
4832:
4828:
4823:
4819:
4815:
4807:
4806:
4802:
4798:
4794:
4776:
4759:
4755:
4750:
4737:
4732:
4729:
4721:
4718:
4715:
4712:
4703:
4700:
4697:
4694:
4688:
4682:
4676:
4668:
4665:
4650:
4644:
4639:
4632:
4627:
4621:
4616:
4613:
4605:
4603:
4598:
4582:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4555:
4550:
4533:
4531:
4527:
4517:
4515:
4494:
4477:
4473:
4463:
4450:
4447:
4440:
4432:
4428:
4421:
4413:
4409:
4402:
4396:
4391:
4387:
4383:
4379:
4375:
4369:
4366:
4363:
4357:
4354:
4346:
4344:
4338:
4334:
4312:
4295:
4291:
4282:
4273:
4260:
4252:
4248:
4244:
4239:
4235:
4228:
4220:
4216:
4209:
4203:
4195:
4182:
4179:
4164:
4161:
4156:
4150:
4140:
4132:
4129:
4125:
4118:
4115:
4111:
4107:
4101:
4093:
4089:
4080:
4079:The function
4077:
4064:
4055:
4049:
4043:
4038:
4035:
4032:
4029:
4026:
4016:
4013:
4010:
4007:
3999:
3993:
3987:
3979:
3976:
3972:
3963:
3959:
3950:
3934:
3917:
3913:
3905:
3900:
3887:
3881:
3877:
3874:
3871:
3867:
3863:
3859:
3855:
3852:
3841:
3835:
3822:
3820:
3816:
3812:
3808:
3804:
3799:
3786:
3778:
3771:
3767:
3760:
3754:
3751:
3745:
3739:
3734:
3723:
3720:
3712:
3702:
3698:
3694:
3691:
3684:
3678:
3669:
3665:
3661:
3657:
3642:
3640:
3636:
3632:
3628:
3615:
3610:
3599:
3596:
3591:
3587:
3580:
3575:
3559:
3557:
3556:eigenfunction
3548:
3528:
3502:
3498:
3494:
3488:
3480:
3476:
3472:
3469:
3460:
3434:
3425:
3422:
3419:
3409:
3406:
3400:
3396:
3392:
3379:
3375:
3365:
3361:
3357:
3351:
3343:
3339:
3335:
3332:
3326:
3316:
3314:
3309:
3296:
3293:
3290:
3284:
3280:
3266:
3258:
3248:
3235:
3227:
3217:
3210:
3207:
3204:
3201:
3190:
3186:
3178:
3174:
3168:
3164:
3155:
3154:
3150:
3123:
3120:
3116:
3109:
3103:
3097:
3089:
3085:
3076:
3072:
3063:
3059:
3041:
3037:
3033:
3028:
3023:
3019:
3015:
3012:
3003:
2990:
2987:
2984:
2978:
2974:
2967:
2959:
2948:
2936:
2930:
2927:
2921:
2915:
2907:
2904:
2891:
2888:
2882:
2879:
2871:
2868:
2865:
2859:
2847:
2836:
2833:
2820:
2814:
2811:
2808:
2805:
2800:
2797:
2794:
2791:
2785:
2779:
2771:
2758:
2756:
2752:
2748:
2744:
2739:
2726:
2723:
2715:
2711:
2707:
2701:
2697:
2691:
2688:
2685:
2677:
2673:
2664:
2662:
2658:
2654:
2650:
2646:
2642:
2637:
2624:
2621:
2618:
2614:
2611:
2606:
2602:
2594:
2588:
2575:
2571:
2567:
2564:
2561:
2557:
2554:
2545:
2532:
2527:
2519:
2516:
2513:
2510:
2507:
2502:
2497:
2489:
2486:
2475:
2471:
2468:in SU(2) and
2467:
2463:
2460:
2456:
2438:
2420:
2407:
2402:
2394:
2391:
2383:
2378:
2370:
2365:
2360:
2352:
2347:
2342:
2329:
2325:
2321:
2318:
2307:
2306:
2301:
2288:
2285:
2281:
2278:
2274:
2271:
2265:
2262:
2258:
2254:
2251:
2248:
2240:
2237:
2223:
2217:
2213:
2209:
2206:
2201:
2197:
2193:
2190:
2185:
2181:
2177:
2173:
2167:
2164:
2160:
2156:
2151:
2147:
2143:
2135:
2134:
2130:
2114:
2111:
2107:
2103:
2100:
2095:
2078:
2074:
2069:
2056:
2051:
2048:
2040:
2037:
2034:
2031:
2022:
2019:
2016:
2013:
2007:
2001:
1995:
1973:
1967:
1962:
1955:
1950:
1944:
1939:
1936:
1928:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1873:
1868:
1851:
1847:
1843:
1833:
1831:
1819:
1814:
1801:
1795:
1792:
1789:
1783:
1777:
1771:
1765:
1757:
1753:
1744:
1740:
1736:
1731:
1718:
1712:
1709:
1706:
1700:
1687:
1681:
1671:
1667:
1663:
1658:
1650:
1627:
1621:
1616:
1613:
1609:
1602:
1599:
1593:
1587:
1581:
1575:
1567:
1563:
1544:
1538:
1532:
1526:
1521:
1517:
1513:
1505:
1496:
1490:
1487:
1481:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1446:
1438:
1434:
1430:
1426:
1422:
1418:
1408:
1395:
1394:
1390:
1386:
1382:
1381:
1380:
1374:
1370:
1366:
1362:
1358:
1342:
1336:
1328:
1324:
1319:
1312:
1304:
1300:
1296:
1293:
1286:
1282:
1278:
1274:
1254:
1251:
1247:
1242:
1237:
1233:
1229:
1225:
1206:
1200:
1195:
1191:
1187:
1179:
1171:
1168:
1160:
1154:
1151:
1144:
1141:
1132:
1128:
1125:
1121:
1117:
1113:
1110:
1106:
1102:
1099:
1095:
1091:
1087:
1083:
1079:
1076:
1072:
1071:
1070:
1049:
1046:
1043:
1037:
1034:
1026:
1023:
1011:
1005:
1001:
997:
991:
983:
979:
971:
970:
967:
966:
965:
963:
960:are given by
959:
950:
948:
944:
916:
912:
908:
904:
897:
893:
877:
874:
871:
864:
856:
852:
848:
842:
836:
831:
827:
823:
814:
808:
800:
796:
787:
779:
776:
772:
739:
737:
733:
729:
725:
721:
717:
710:
706:
705:
700:
696:
692:
687:
685:
657:
653:
646:
642:
638:
634:
630:
626:
619:
615:
611:
607:
600:
572:
568:
564:
529:
525:
521:
517:
512:
486:
482:
454:
451:
447:
420:
416:
412:
408:
404:
400:
396:
392:
388:
385:
384:Hilbert space
381:
377:
373:
369:
365:
358:
355:
354:Hecke algebra
351:
347:
343:
339:
336:
332:
326:
316:
314:
310:
306:
302:
298:
294:
290:
285:
283:
279:
275:
269:
267:
263:
259:
255:
251:
247:
243:
240:and proposed
235:
232:
228:
224:
220:
216:
212:
204:
200:
197:. Since when
196:
192:
188:
185:, a class of
184:
181:
177:
173:
170:
166:
162:
158:
154:
150:
146:
143:
140:
130:
128:
123:
121:
117:
113:
109:
108:prior results
105:
99:
95:
90:
86:
82:
78:
75:
71:
67:
63:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
27294:Balanced set
27268:Distribution
27206:Applications
27059:Krein–Milman
27044:Closed graph
26842:
26838:
26820:
26814:
26805:
26801:
26772:
26766:
26754:
26736:
26718:, Springer,
26715:
26694:
26690:
26662:
26641:
26621:
26617:
26613:
26592:
26572:
26560:
26542:
26536:
26506:
26502:
26474:
26468:
26436:
26430:
26386:
26382:
26330:
26326:
26315:
26300:
26280:
26253:
26249:
26232:
26228:
26208:
26202:
26177:
26161:
26157:
26139:
26118:
26099:
26095:
26067:
26063:
26045:
26041:
26020:
26008:
25996:
25985:
25973:
25960:
25948:
25936:
25924:
25912:
25900:
25888:
25876:
25864:
25852:
25840:
25828:
25816:
25804:
25792:
25780:
25769:
25758:Stade (1999)
25742:
25738:
25730:
25723:
25717:
25711:
25707:
25700:
25696:
25692:
25680:
25668:
25656:
25587:
25582:
25571:
25559:
25547:
25538:
25527:
25516:
25511:, p. 46
25504:
25499:, p. 41
25492:
25481:
25470:
25449:, p. 38
25442:
25431:
25420:
25403:
25371:
25367:
25362:
25357:, section 21
25350:
25339:
25298:
25287:
25276:
25265:
25253:
25235:Anker (1991)
25232:
25190:
25134:
24930:
24927:
24893:
24889:
24887:
24673:
24619:
24615:
24611:
24607:
24487:
24482:
24478:
24470:
24462:
24344:
24340:
24338:
24332:
24320:
24313:
24308:
24304:
24300:
24296:
24294:
24165:
24160:
24156:
24152:
24148:
24144:
24138:
24127:
24123:
24119:
24114:The general
24113:
23651:
23618:
23614:
23610:
23605:
23601:
23597:
23593:
23551:
23540:
23536:
23531:
23527:
23525:
23313:-functions:
23310:
23306:
23302:
23298:
23210:
23208:
23146:
23141:
23059:
23055:
23051:
23047:
23043:
23039:
23037:
22827:
22604:
22600:
22596:
22592:
22588:
22584:
22580:
22576:
22541:
22537:
22533:
22529:
22525:
22521:
22517:
22513:
22508:
22504:
22500:
22492:
22488:
22480:
22475:
22471:
22467:
22463:
22457:
22436:
22434:
22297:
22295:
22222:
22193:
22023:
22021:
21923:
21919:
21915:
21907:
21903:
21899:
21895:
21891:
21822:
21815:
21808:
21804:
21802:
21747:
21739:
21733:
21728:
21693:
21562:
21558:
21556:
21475:
21468:
21462:
21433:
21429:
21425:
21421:
21414:
21410:
21368:
21366:
21286:
21279:
21168:
21164:
21160:
21158:
21129:
21125:
21027:
21019:
21015:
20842:
20836:
20832:
20670:
20578:
20577:
20453:
20449:
20445:
20417:
20413:
20409:
20405:
20401:
20376:, and hence
20373:
20369:
20365:
20361:
20357:
20353:
20351:
20093:
20091:
20018:) = 1 where
20015:
20011:
20004:
19606:
19603:
19461:states that
19458:
19456:
19357:
19353:
19349:
19341:
19304:
19295:
19079:Killing form
19074:
19070:
19042:
19039:multiplicity
19038:
18984:
18939:
18825:
18743:
18665:
18619:
18519:
18491:
18489:
18475:
18445:
18305:
18297:
18289:
18287:
17735:
17731:
17727:
17723:
17719:
17715:
17711:
17709:
17625:
17509:
17458:
17452:
17391:
17271:
17267:
17265:
17101:
17097:
17038:
17036:
16987:
16983:
16972:
16965:
16963:
16955:
16951:
16947:
16943:
16939:
16935:
16933:
16797:
16717:
16713:
16709:
16705:
16698:
16691:
16684:
16677:
16670:
16663:
16656:
16652:
16648:
16644:
16642:
16572:
16491:
16448:proved that
16438:
16434:
16427:
16420:
16413:
16406:
16399:
16395:
16388:
16381:
16377:
16373:
16369:
16362:
16358:
16354:
16343:
16341:
16333:
16208:
16203:
16197:
16080:
16078:is given by
16075:
16067:
16063:
16059:
16019:
16011:
16007:
16003:
15999:
15995:
15992:compact dual
15991:
15987:
15983:
15978:
15917:is just the
15685:Weyl chamber
15646:
15384:
15381:
15288:
15286:is given by
15283:
15279:
15277:
15272:
15268:
15264:
15256:
15254:
15158:
15109:
15107:
15067:
14917:
14912:
14908:
14904:
14902:
14897:
14869:
14812:
14810:
14667:
14592:
14453:
14449:Hermann Weyl
14413:
14407:
14359:
14352:
14283:
14251:
14243:
14141:
14133:
14131:
14091:
14086:
14078:
14069:
14065:
14059:
13895:
13887:
13883:
13879:
13875:
13873:
13778:
13775:
13694:
13689:
13685:
13681:
13677:
13670:
13637:
13559:
13554:
13550:
13546:
13544:
13247:
13244:
13128:
13090:
13088:
12791:
12788:
12665:
12656:
12652:
12650:
12458:
12455:
12357:
12352:
12350:
12173:
12003:is given by
11998:
11844:
11840:
11836:
11832:
11828:
11826:
11740:
11543:
11540:
11472:
11276:
11190:
11118:
11114:
11110:
11106:
11102:
11098:
11094:
11090:
11083:
11081:
10947:
10877:
10873:
10869:
10864:
10860:
10856:
10852:
10850:
10848:over SU(2).
10845:
10839:
10835:
10831:
10824:
10822:
10746:
10739:
10735:
10654:
10650:
10646:
10642:
10640:
10511:
10326:
10321:
10317:
10313:
10309:
10307:
10231:
10228:
10158:
10153:
10149:
10145:
10141:
10137:
10135:
9937:
9932:
9928:
9924:
9920:
9916:
9914:
9764:
9761:
9649:
9646:
9552:
9545:
9508:
9504:
9500:
9496:
9492:
9488:
9484:
9482:
9322:
9317:
9313:
9309:
9305:
9301:
9297:
9293:
9289:
9285:
9281:
9277:
9273:
9269:
9265:
9261:
9257:
9253:
9248:
9244:
9167:
8928:
8787:
8783:
8779:
8777:
8674:
8672:of matrices
8666:
8662:
8627:
8625:
8566:
8449:
8279:
8188:
8185:
8086:
8083:
7956:
7920:
7916:
7908:
7906:
7838:
7836:
7761:
7756:
7748:
7742:
7581:
7576:
7572:
7567:
7478:
7467:
7276:
7274:) / SO(2)):
7271:
7260:
7256:
7255:As for SL(2,
7254:
7220:
7075:
7070:
7066:
7065:, the SL(2,
7062:
7055:
7051:
7049:
6962:
6959:
6888:
6854:
6800:
6797:
6645:
6640:
6637:power series
6628:
6622:
6549:
6544:
6540:
6536:
6487:
6352:
6207:
6105:
6101:
5966:
5962:
5955:
5954:The adjoint
5953:
5818:
5815:
5694:
5686:
5684:
5595:
5590:
5584:
5577:
5569:
5567:
5530:
5527:
5432:
5429:
5335:
5299:
5295:
5291:
5289:
5280:
5268:
5251:
5158:
5088:
5084:
5080:
5076:
5073:
5069:
5036:
4957:
4951:
4897:
4894:
4808:
4800:
4796:
4792:
4757:
4753:
4751:
4669:
4666:
4606:
4599:
4534:
4529:
4525:
4523:
4464:
4347:
4342:
4336:
4332:
4280:
4274:
4183:
4180:
4081:
4078:
3951:
3915:
3911:
3903:
3901:
3823:
3818:
3810:
3800:
3643:
3638:
3634:
3629:
3560:
3549:
3461:
3317:
3312:
3310:
3156:
3152:
3148:
3074:
3070:
3061:
3057:
3004:
2908:
2905:
2837:
2834:
2759:
2754:
2750:
2740:
2665:
2660:
2656:
2652:
2648:
2644:
2640:
2638:
2546:
2473:
2469:
2465:
2461:
2458:
2454:
2421:
2308:
2302:
2241:
2238:
2136:
2128:
2076:
2072:
2070:
1850:quaternionic
1845:
1841:
1839:
1829:
1815:
1742:
1738:
1734:
1732:
1565:
1561:
1471:
1467:
1463:
1459:
1455:
1451:
1444:
1436:
1432:
1424:
1420:
1416:
1410:
1392:
1388:
1384:
1383:
1378:
1364:
1360:
1284:
1276:
1272:
1249:
1240:
1235:
1231:
1227:
1223:
1130:
1126:
1115:
1108:
1104:
1100:
1093:
1089:
1085:
1081:
1074:
1068:
957:
951:
946:
943:Haar measure
914:
910:
906:
902:
895:
891:
785:
770:
740:
735:
731:
727:
723:
719:
715:
708:
702:
698:
694:
690:
688:
655:
651:
644:
640:
636:
617:
609:
605:
598:
570:
566:
527:
523:
518:denotes the
515:
513:
480:
452:
444:, so by the
406:
402:
398:
394:
390:
386:
379:
375:
371:
367:
363:
356:
349:
341:
330:
328:
312:
300:
296:
286:
270:
261:
257:
249:
245:
241:
233:
230:
223:Hermann Weyl
198:
194:
190:
171:
164:
160:
151:and complex
144:
136:
124:
97:
93:
76:
21:
15:
27223:Heat kernel
27213:Hardy space
27120:Trace class
27034:Hahn–Banach
26996:Topological
26845:: 289–433,
26712:Lang, Serge
26638:Lang, Serge
26624:: 273–284,
26235:: 252–255,
26164:: 106–146,
25425:Davies 1989
25193:-invariant
23144:-function:
19186:root system
18448:root system
16796:and define
11839:, the SL(2,
10880:defined by
9170:hyperboloid
4803:-invariant
3151:yields the
2131:-invariant
1429:centraliser
1369:centraliser
913:), where π(
415:commutative
401:). Because
18:mathematics
27347:Categories
27156:C*-algebra
26971:Properties
26102:: 97–205,
26064:Math. Ann.
26034:References
26013:Anker 1991
25905:Knapp 2001
25893:Knapp 2001
25881:Knapp 2001
25475:Anker 1991
25415:is unique.
24622:such that
22470:-function
21738:and hence
19190:Weyl group
19105:and hence
19069:, so that
18577:, so that
14817:Weyl group
11113:, so that
9507:, so that
9268:) leaving
9168:So on the
5816:satisfies
4524:The group
3315:: the map
1840:The group
1405:See also:
1357:normaliser
1281:Weyl group
1279:is in the
485:characters
419:C* algebra
335:semisimple
231:c-function
139:unimodular
27130:Unbounded
27125:Transpose
27083:Operators
27012:Separable
27007:Reflexive
26992:Algebraic
26978:Barrelled
26545:: 1–154,
26525:125806386
26509:: 1–111,
26270:120718983
26223:(1962),
26211:: 609–612
26084:123060173
25622:−
25614:λ
25603:λ
25599:χ
25532:Loeb 1979
25240:seminorms
25214:∗
25162:∗
25113:∞
25074:Δ
24987:∞
24979:∈
24951:∖
24871:λ
24859:−
24846:λ
24830:λ
24826:φ
24819:λ
24810:~
24797:∗
24779:∫
24725:−
24705:∫
24594:λ
24582:−
24569:λ
24552:λ
24543:~
24530:∗
24512:∫
24447:∗
24425:for μ in
24410:μ
24399:∗
24368:∗
24318:pg. 37).
24271:λ
24268:
24244:−
24231:λ
24204:≤
24193:λ
24184:~
24081:α
24065:α
24036:Γ
24029:α
24005:α
23937:α
23930:λ
23916:α
23900:α
23861:Γ
23839:α
23832:λ
23812:α
23773:Γ
23756:α
23749:λ
23737:Γ
23723:α
23716:λ
23707:−
23680:λ
23670:α
23600:= 1, and
23574:α
23571:±
23493:ℓ
23471:ℓ
23439:ℓ
23431:provided
23416:λ
23390:λ
23351:λ
23272:ξ
23265:λ
23240:ξ
23233:λ
23189:λ
23160:λ
23126:∗
23109:−
23087:∗
23008:ℓ
22986:ℓ
22954:ℓ
22946:whenever
22928:λ
22903:λ
22865:λ
22808:λ
22782:λ
22775:π
22757:λ
22753:π
22746:λ
22673:−
22665:∩
22653:σ
22649:∫
22627:λ
22561:∗
22398:¯
22381:λ
22363:σ
22359:∫
22349:λ
22268:−
22242:∞
22239:→
22159:−
22131:λ
22124:¯
22107:λ
22087:σ
22083:∫
22077:ρ
22074:−
22071:λ
22039:λ
22035:φ
21993:−
21972:λ
21954:∫
21936:λ
21932:φ
21864:−
21772:∈
21765:⋃
21669:ρ
21666:−
21663:λ
21646:λ
21637:∼
21613:λ
21609:φ
21600:, since
21538:λ
21524:λ
21507:∈
21500:∑
21491:λ
21487:φ
21339:−
21319:∑
21306:−
21300:
21261:μ
21258:−
21247:λ
21239:μ
21229:Λ
21226:∈
21223:μ
21219:∑
21213:ρ
21210:−
21207:λ
21191:λ
21110:λ
21106:φ
21087:ρ
21066:λ
21048:λ
21044:φ
21040:Δ
20997:α
20981:α
20977:∑
20973:−
20962:Δ
20923:α
20909:−
20906:α
20903:
20891:α
20875:α
20871:∑
20867:−
20780:∑
20777:−
20763:Δ
20733:α
20716:α
20652:α
20643:α
20640:
20631:α
20615:α
20611:∑
20607:−
20596:Δ
20329:~
20289:∖
20270:∈
20244:λ
20232:−
20219:λ
20200:∗
20170:→
20153:∖
20122:~
20113:↦
20075:α
20070:α
20054:Σ
20050:∈
20047:α
20043:∑
20026:ρ
19991:α
19983:−
19975:α
19969:α
19954:α
19911:α
19904:λ
19890:α
19874:α
19835:Γ
19813:α
19806:λ
19786:α
19747:Γ
19730:α
19723:λ
19711:Γ
19697:α
19690:λ
19681:−
19657:Σ
19653:∈
19650:α
19646:∏
19642:⋅
19623:λ
19587:λ
19575:−
19562:λ
19536:λ
19532:φ
19525:λ
19516:~
19503:∗
19485:∫
19422:λ
19419:−
19415:φ
19393:∫
19383:λ
19374:~
19326:∗
19280:∗
19146:∗
19023:α
19011:
19000:α
18962:≠
18957:α
18910:∈
18887:α
18856:∈
18842:α
18809:∗
18724:⋅
18714:
18649:−
18539:±
18423:α
18417:α
18406:λ
18400:α
18394:π
18388:
18374:α
18370:∏
18366:⋅
18358:−
18350:λ
18338:⋅
18326:λ
18312:, giving
18296:for SL(2,
18271:λ
18249:λ
18205:λ
18183:λ
18179:φ
18172:λ
18163:~
18150:∗
18132:∫
18125:λ
18103:λ
18051:λ
18044:Φ
18038:∗
18027:λ
18015:~
18002:∗
17984:∫
17963:∗
17920:λ
17898:λ
17881:⋅
17870:λ
17856:⋅
17816:λ
17812:φ
17805:λ
17796:~
17783:∗
17765:∫
17690:λ
17678:~
17663:λ
17655:∼
17642:∗
17591:λ
17584:Φ
17574:∫
17556:λ
17549:Φ
17543:∗
17529:λ
17490:λ
17483:Φ
17477:∗
17434:∗
17426:Δ
17411:Δ
17405:∗
17367:∗
17356:⋅
17322:∫
17312:⋅
17283:∫
17221:∫
17193:∗
17140:∖
17122:∞
17096:, define
17070:∖
17059:∞
17011:Δ
17004:Δ
16978:and let Δ
16882:∫
16829:∖
16818:∞
16756:∖
16738:∞
16591:∖
16532:∖
16307:λ
16292:λ
16270:λ
16266:χ
16256:λ
16247:~
16236:λ
16232:∑
16171:λ
16159:¯
16144:λ
16140:χ
16113:∫
16103:λ
16094:~
16044:∗
15948:δ
15902:λ
15893:~
15882:¯
15875:λ
15832:λ
15818:σ
15791:∈
15788:σ
15784:∑
15759:λ
15755:Φ
15735:δ
15729:λ
15669:∗
15630:λ
15607:λ
15582:λ
15578:Φ
15571:λ
15562:~
15549:∗
15531:∫
15524:λ
15501:λ
15476:λ
15472:Φ
15465:λ
15456:~
15443:∗
15430:∫
15350:λ
15347:−
15343:Φ
15321:∫
15311:λ
15302:~
15232:λ
15203:λ
15199:χ
15171:λ
15167:Φ
15142:∗
15016:δ
14980:∫
14926:∫
14775:δ
14762:σ
14756:λ
14742:σ
14715:∈
14712:σ
14708:∑
14680:λ
14676:χ
14651:∗
14615:×
14610:∗
14574:λ
14570:π
14566:
14554:λ
14531:∈
14501:λ
14497:π
14493:
14466:λ
14462:χ
14433:∗
14376:⋅
14326:
14300:
14217:⊕
14068:,1) with
14038:λ
14021:λ
14018:π
14008:
13996:π
13993:λ
13976:λ
13972:φ
13965:λ
13956:~
13945:∞
13936:∫
13924:π
13846:λ
13842:φ
13838:⋅
13832:λ
13823:~
13808:λ
13793:~
13756:ξ
13749:λ
13740:~
13722:ξ
13707:λ
13703:π
13590:∫
13557:) define
13528:λ
13511:λ
13508:π
13498:
13486:π
13483:λ
13474:λ
13465:~
13454:∞
13449:∞
13446:−
13442:∫
13429:π
13396:
13384:
13365:λ
13362:
13351:∞
13346:∞
13343:−
13339:∫
13335:λ
13328:λ
13322:λ
13313:~
13302:∞
13293:∫
13280:π
13228:λ
13215:λ
13209:
13200:λ
13188:~
13177:∞
13168:∫
13162:π
13145:
13107:~
13060:π
12997:∞
12988:∫
12983:π
12900:∞
12895:∞
12892:−
12888:∫
12882:∞
12877:∞
12874:−
12870:∫
12812:∞
12807:∞
12804:−
12800:∫
12721:∞
12716:∞
12713:−
12709:∫
12702:π
12691:−
12625:λ
12616:−
12602:
12588:∞
12579:∫
12563:λ
12557:−
12534:−
12500:∞
12491:∫
12481:λ
12472:~
12393:∞
12388:∞
12385:−
12381:∫
12310:λ
12304:−
12268:−
12234:∞
12225:∫
12219:∞
12214:∞
12211:−
12207:∫
12197:λ
12188:~
12137:α
12115:∫
12105:λ
12096:~
12044:α
12034:∫
12016:λ
12012:φ
11977:λ
11960:λ
11957:π
11947:
11935:π
11932:λ
11922:λ
11913:~
11893:λ
11889:φ
11883:∞
11878:∞
11875:−
11871:∫
11807:λ
11795:~
11780:λ
11772:∼
11759:∗
11722:λ
11718:φ
11706:λ
11699:Φ
11678:λ
11675:π
11672:
11666:π
11649:λ
11642:Φ
11636:∗
11622:λ
11594:λ
11590:φ
11583:λ
11569:λ
11562:Φ
11556:∗
11522:∗
11508:Δ
11492:Δ
11486:∗
11443:∗
11434:⋅
11415:∫
11398:⋅
11370:∫
11338:∗
11327:⋅
11312:⋅
11292:∗
11251:∗
11230:∫
11200:∫
11134:∗
10989:∫
10959:∫
10917:∘
10898:∫
10797:Δ
10758:Δ
10710:Δ
10678:−
10669:Δ
10618:∂
10602:
10579:∂
10566:∂
10556:
10544:
10521:∂
10486:∂
10476:
10464:
10438:∂
10422:∂
10403:∂
10396:
10373:∂
10341:Δ
10337:−
10274:−
10265:Δ
10240:Δ
10109:
10089:∞
10084:∞
10081:−
10077:∫
10073:π
10041:∫
10021:
9994:∞
9989:∞
9986:−
9982:∫
9978:π
9946:∫
9891:∂
9875:∂
9859:∂
9822:∂
9804:∂
9786:∂
9742:∂
9726:∂
9689:∂
9671:∂
9613:−
9604:−
9586:−
9573:−
9561:Δ
9528:
9439:−
9363:−
9140:−
9131:
9113:
9093:
9075:
9060:−
9041:
9023:
8988:
8970:
8947:⋅
8910:∗
8890:⋅
8856:−
8842:−
8750:
8739:
8730:−
8720:
8706:
8531:ξ
8516:λ
8512:π
8482:∫
8466:λ
8435:λ
8418:λ
8415:π
8405:
8393:π
8390:λ
8359:×
8353:∞
8328:→
8254:ξ
8241:ξ
8226:λ
8222:π
8201:λ
8197:Φ
8167:−
8121:−
8117:π
8095:ξ
8049:ξ
8044:λ
8038:−
8032:−
7993:ξ
7982:−
7969:λ
7965:π
7919:), where
7889:~
7875:∼
7862:∗
7813:λ
7810:ℑ
7791:≤
7780:λ
7717:∞
7691:φ
7665:Δ
7549:λ
7545:φ
7515:λ
7501:λ
7497:φ
7487:Δ
7451:λ
7434:λ
7431:π
7421:
7409:π
7406:λ
7398:¯
7393:~
7367:~
7355:∞
7350:∞
7347:−
7343:∫
7327:¯
7285:∫
7204:λ
7187:λ
7184:π
7174:
7162:π
7159:λ
7150:λ
7141:~
7121:λ
7117:φ
7111:∞
7106:∞
7103:−
7099:∫
7030:λ
7021:~
7006:λ
6998:∼
6985:∗
6934:λ
6931:−
6927:φ
6920:∫
6911:λ
6902:~
6836:λ
6832:ϕ
6813:λ
6809:Φ
6781:θ
6772:λ
6766:−
6760:−
6751:θ
6748:
6739:
6733:−
6727:
6713:π
6701:∫
6694:π
6658:λ
6654:φ
6633:Wronskian
6607:λ
6603:Φ
6584:−
6576:λ
6562:λ
6558:ϕ
6520:
6504:λ
6465:λ
6462:π
6456:
6448:λ
6445:π
6424:
6418:λ
6410:λ
6407:
6398:∫
6381:λ
6367:λ
6334:λ
6300:λ
6286:λ
6272:Δ
6249:λ
6245:Φ
6239:∗
6221:λ
6189:∗
6150:Δ
6132:Δ
6126:∗
6073:∗
6057:⋅
6034:∫
6017:⋅
5978:∫
5922:∗
5906:⋅
5883:∫
5866:⋅
5827:∫
5769:∞
5764:∞
5761:−
5757:∫
5712:∗
5642:Δ
5604:Δ
5482:Δ
5441:Δ
5398:⋅
5221:
5196:∞
5191:∞
5188:−
5184:∫
5167:∫
5136:∂
5129:
5123:−
5109:∂
5105:−
5099:Δ
5006:∂
4988:∂
4971:−
4965:Δ
4919:−
4837:−
4730:−
4574:∣
4429:ξ
4414:λ
4410:π
4380:∫
4364:λ
4335:([0,∞) ×
4249:ξ
4236:ξ
4221:λ
4217:π
4196:λ
4192:Φ
4162:−
4116:−
4112:π
4090:ξ
4044:ξ
4039:λ
4033:−
4027:−
3988:ξ
3977:−
3964:λ
3960:π
3914:), where
3878:λ
3875:
3853:≤
3842:λ
3775:∞
3755:
3727:Δ
3611:λ
3607:Φ
3588:λ
3576:λ
3572:Φ
3568:Δ
3532:~
3492:∖
3473:∈
3438:~
3429:⟼
3410:λ
3397:λ
3372:→
3355:∖
3294:λ
3281:λ
3274:¯
3267:λ
3252:~
3236:λ
3221:~
3211:∫
3196:¯
3165:∫
3133:¯
3121:−
3090:∗
3034:⋆
3029:∗
2988:λ
2975:λ
2960:λ
2956:Φ
2949:λ
2940:~
2931:∫
2883:λ
2880:−
2876:Φ
2869:∫
2860:λ
2851:~
2812:
2806:λ
2798:λ
2795:
2772:λ
2768:Φ
2692:−
2683:Δ
2678:∗
2612:
2584:∞
2579:∞
2576:−
2572:∫
2555:∫
2524:∂
2517:
2508:−
2494:∂
2490:−
2484:Δ
2399:∂
2375:∂
2357:∂
2339:∂
2322:−
2316:Δ
2263:−
2165:−
2049:−
1901:∣
1778:π
1758:λ
1754:χ
1716:∞
1668:∫
1655:‖
1648:‖
1614:−
1576:π
1527:λ
1500:Δ
1238:, where Δ
1201:λ
1165:Δ
1142:λ
1120:character
1035:−
1016:′
1012:λ
1002:∫
984:λ
980:φ
857:λ
853:φ
849:⋅
828:∫
809:π
801:λ
797:χ
631:μ on the
547:′
520:commutant
338:Lie group
276:" due to
254:real rank
225:from the
27332:Category
27144:Algebras
27026:Theorems
26983:Complete
26952:Schwartz
26898:glossary
26714:(1998),
26682:(1969),
26661:(2001),
26640:(2001),
26423:16589101
26375:16589034
26299:(1977),
26279:(1957),
26138:(1978),
26117:(1989),
26070:: 1–23,
25647:form an
23637:for the
22587:, where
22384:′
22134:′
22110:′
21975:′
21090:‖
21084:‖
21069:‖
21063:‖
21014:so that
16433:and let
16006:, where
15858:so that
13776:so that
13553:in SL(2,
13006:′
12909:′
12821:′
12789:In fact
12730:′
12140:′
12082:so that
12047:′
10834:, where
6639:in sinh
4667:acts as
1988:acts as
1275:, where
1145:′
1092:) where
643:between
450:spectrum
176:Godement
27135:Unitary
27115:Nuclear
27100:Compact
27095:Bounded
27090:Adjoint
27064:Min–max
26957:Sobolev
26942:Nuclear
26932:Hilbert
26927:Fréchet
26892: (
26808:: 47–87
26791:2041084
26716:SL(2,R)
26491:2372772
26461:0094407
26453:2372786
26414:1063558
26366:1063477
26335:Bibcode
26241:0150239
26108:0084713
23592:. Then
23293:where ξ
22524:. Once
21561:(λ) is
16990:. Then
16683:. Thus
15990:has as
14815:is the
11827:Taking
9487:equals
7050:Taking
5528:where Δ
5079:) with
4528:= SL(2,
3801:By the
3005:Taking
2741:By the
2659:) sinh
2464:) with
1844:= SL(2,
1448:induced
1427:is the
1367:by its
1244:is the
1122:of the
1084:* =Hom(
629:measure
530:, then
483:, i.e.
149:SL(2,R)
133:History
87:of the
40:of the
27110:Normal
26947:Orlicz
26937:Hölder
26917:Banach
26906:Spaces
26894:topics
26789:
26743:
26722:
26669:
26648:
26599:
26581:
26523:
26489:
26459:
26451:
26421:
26411:
26403:
26373:
26363:
26355:
26307:
26268:
26239:
26184:
26146:
26125:
26106:
26082:
25706:where
23974:where
22599:. The
22223:Since
21803:where
21285:under
21159:Since
20940:where
20669:where
20456:is in
20352:Since
20134:
19602:where
19294:. The
19188:. Its
19045:= exp
19041:. Let
17935:since
16716:. Let
16573:Since
16490:where
16405:. Let
15715:since
15647:where
15068:where
14868:and δ(
14811:where
13874:where
13245:Hence
11082:Since
9647:where
9483:where
7472:is an
7270:(SL(2,
3554:is an
3077:) and
1745:) and
1640:where
635:space
573:. Now
352:. The
112:Mehler
20:, the
26922:Besov
26787:JSTOR
26687:(PDF)
26521:S2CID
26487:JSTOR
26449:JSTOR
26405:88737
26401:JSTOR
26357:88521
26353:JSTOR
26266:S2CID
26080:S2CID
25246:Notes
24131:(λ).
23635:Gauss
23101:onto
22455:and π
21856:onto
21557:Here
20843:Thus
19945:with
17266:Then
16954:with
16349:be a
16070:of a
15683:is a
14246:be a
13669:with
12174:Thus
11608:with
10229:Thus
9316:) on
9308:and (
9304:) in
9284:with
8929:Thus
8614:as a
6960:then
6539:and ±
5430:then
4512:as a
4330:onto
3056:with
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27270:(or
26988:Dual
26741:ISBN
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26419:PMID
26371:PMID
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23209:The
23050:and
22583:' /
22462:for
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18923:}
18920:)
18915:a
18907:H
18904:(
18899:X
18896:)
18893:H
18890:(
18884:=
18881:]
18878:X
18875:,
18872:H
18869:[
18866::
18861:g
18853:X
18850:{
18847:=
18836:g
18803:a
18778:p
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18730:.
18727:K
18719:p
18708:=
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18680:=
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18426:)
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18414:(
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18347:2
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18332:=
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18306:b
18298:R
18290:b
18274:.
18268:d
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18175:)
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18111:|
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18100:2
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18090:|
18084:A
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18034:M
18030:)
18024:2
18021:(
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17997:+
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17980:=
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17974:g
17971:(
17968:F
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17955:=
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17949:g
17946:(
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17923:,
17917:d
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17901:)
17895:2
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17660:(
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17638:M
17634:(
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17602:)
17599:k
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17570:=
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17564:1
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17506:λ
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17217:=
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17077:/
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17064:(
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17050:C
17039:F
17023:.
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17007:=
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16988:U
16986:/
16984:G
16980:c
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16973:K
16971:/
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16878:=
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16843:)
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16836:/
16832:G
16826:U
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16669:/
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16660:0
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16627:+
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16605:/
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16586:0
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16559:.
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16546:=
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16539:/
16535:G
16529:K
16507:a
16495:+
16492:A
16478:,
16475:U
16470:+
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16459:=
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16442:0
16439:P
16435:A
16431:0
16428:G
16424:0
16421:P
16419:·
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16414:K
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16320:.
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16301:d
16295:)
16289:(
16286:d
16281:)
16278:u
16275:(
16259:)
16253:(
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16228:=
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16219:(
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16184:u
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16174:)
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16165:d
16155:)
16152:u
16149:(
16131:)
16128:u
16125:(
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16109:=
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16076:U
16068:f
16064:T
16060:A
16039:+
16033:a
16020:d
16018:/
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16000:U
15996:U
15988:U
15986:/
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15951:(
15945:)
15940:X
15936:e
15932:(
15929:F
15905:)
15899:(
15890:F
15878:)
15872:(
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15841:)
15838:X
15835:(
15829:i
15825:e
15821:)
15815:(
15810:n
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15801:s
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15780:=
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15772:X
15768:e
15764:(
15751:)
15746:X
15742:e
15738:(
15732:)
15726:(
15723:d
15701:a
15664:+
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15633:,
15626:d
15620:2
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15610:)
15604:(
15601:d
15597:|
15593:)
15590:g
15587:(
15574:)
15568:(
15559:F
15544:+
15538:a
15527:=
15520:d
15514:2
15509:|
15504:)
15498:(
15495:d
15491:|
15487:)
15484:g
15481:(
15468:)
15462:(
15453:F
15437:a
15422:|
15418:W
15414:|
15409:1
15404:=
15401:)
15398:g
15395:(
15392:F
15368:g
15365:d
15361:)
15358:g
15355:(
15339:)
15336:g
15333:(
15330:F
15325:G
15317:=
15314:)
15308:(
15299:F
15284:F
15280:U
15273:T
15271:/
15269:U
15265:G
15257:G
15241:.
15235:)
15229:(
15226:d
15221:)
15216:X
15212:e
15208:(
15192:=
15189:)
15184:X
15180:e
15176:(
15136:t
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15127:=
15122:a
15110:G
15091:t
15086:i
15083:=
15078:a
15054:.
15051:X
15048:d
15042:2
15037:|
15032:)
15027:X
15023:e
15019:(
15012:|
15007:)
15002:X
14998:e
14994:(
14991:F
14985:a
14972:|
14968:W
14964:|
14959:1
14954:=
14951:g
14948:d
14944:)
14941:g
14938:(
14935:F
14930:G
14913:F
14909:U
14905:G
14898:d
14882:t
14870:e
14856:T
14852:/
14848:)
14845:T
14842:(
14837:U
14833:N
14829:=
14826:W
14813:W
14797:,
14791:)
14786:X
14782:e
14778:(
14768:)
14765:X
14759:(
14753:i
14749:e
14745:)
14739:(
14734:n
14731:g
14728:i
14725:s
14718:W
14701:=
14698:)
14693:X
14689:e
14685:(
14645:t
14620:t
14604:t
14579:.
14560:=
14557:)
14551:(
14548:d
14544:,
14541:)
14536:t
14528:X
14525:(
14522:,
14519:)
14514:X
14510:e
14506:(
14487:=
14484:)
14479:X
14475:e
14471:(
14427:t
14414:U
14410:λ
14394:.
14391:U
14388:A
14385:U
14382:=
14379:U
14373:P
14370:=
14367:G
14339:,
14334:u
14329:i
14320:=
14317:P
14313:,
14308:t
14303:i
14294:=
14291:A
14269:.
14264:t
14252:U
14244:T
14230:.
14225:u
14220:i
14212:u
14207:=
14202:g
14178:u
14154:g
14142:U
14134:G
14122:λ
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14087:C
14079:R
14070:N
14066:N
14041:.
14035:d
14030:)
14025:2
14012:(
14000:2
13987:)
13984:w
13981:(
13968:)
13962:(
13953:f
13940:0
13928:2
13920:1
13915:=
13912:)
13909:w
13906:(
13903:f
13891:1
13888:f
13884:i
13882:(
13880:g
13876:w
13860:,
13857:)
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13851:(
13835:)
13829:(
13820:f
13814:=
13811:)
13805:(
13800:1
13790:f
13760:0
13752:)
13746:(
13737:f
13731:=
13726:0
13718:)
13715:f
13712:(
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13686:i
13684:(
13682:g
13680:(
13678:f
13674:1
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13624:,
13621:k
13618:d
13614:)
13611:w
13608:k
13605:g
13602:(
13599:f
13594:K
13586:=
13583:)
13580:w
13577:(
13572:1
13568:f
13555:R
13551:g
13547:i
13531:.
13525:d
13520:)
13515:2
13502:(
13490:2
13477:)
13471:(
13462:f
13433:2
13425:2
13421:1
13416:=
13413:t
13410:d
13404:2
13401:t
13387:t
13376:2
13372:/
13368:t
13332:d
13325:)
13319:(
13310:f
13297:0
13284:2
13276:2
13272:1
13267:=
13264:)
13261:i
13258:(
13255:f
13231:.
13225:d
13221:)
13218:t
13212:(
13203:)
13197:i
13194:(
13185:f
13172:0
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13154:=
13151:)
13148:t
13139:(
13136:F
13104:f
13091:F
13075:.
13072:)
13069:x
13066:(
13063:f
13057:2
13054:=
13051:r
13048:d
13044:r
13040:)
13034:2
13029:2
13025:r
13019:+
13016:x
13012:(
13003:f
12992:0
12980:2
12976:=
12973:u
12970:d
12966:t
12963:d
12958:)
12952:2
12946:2
12942:u
12938:+
12933:2
12929:t
12922:+
12919:x
12915:(
12906:f
12866:=
12863:t
12860:d
12855:)
12849:2
12844:2
12840:t
12834:+
12831:x
12827:(
12818:F
12775:.
12772:t
12769:d
12764:)
12758:2
12753:2
12749:t
12743:+
12740:x
12736:(
12727:F
12699:2
12694:1
12685:=
12682:)
12679:x
12676:(
12673:f
12657:f
12653:F
12637:.
12634:t
12631:d
12622:t
12619:i
12612:e
12608:)
12605:t
12596:(
12593:F
12583:0
12575:=
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12569:d
12560:i
12553:a
12548:)
12543:2
12537:2
12530:a
12526:+
12521:2
12517:a
12510:(
12505:F
12495:0
12487:=
12484:)
12478:(
12469:f
12442:,
12439:t
12436:d
12431:)
12425:2
12420:2
12416:t
12410:+
12407:u
12403:(
12398:f
12377:=
12374:)
12371:u
12368:(
12365:F
12353:F
12337:,
12334:b
12331:d
12327:a
12324:d
12318:2
12314:/
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12300:a
12295:)
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12280:b
12276:+
12271:2
12264:a
12260:+
12255:2
12251:a
12244:(
12239:f
12229:0
12203:=
12200:)
12194:(
12185:f
12160:,
12157:s
12154:d
12150:)
12147:s
12144:(
12133:)
12130:s
12127:(
12124:f
12119:S
12111:=
12108:)
12102:(
12093:f
12070:,
12067:k
12064:d
12060:)
12057:g
12054:k
12051:(
12038:K
12030:=
12027:)
12024:g
12021:(
12001:λ
11980:,
11974:d
11969:)
11964:2
11951:(
11939:2
11925:)
11919:(
11910:f
11904:)
11901:x
11898:(
11867:=
11864:)
11861:x
11858:(
11855:f
11845:F
11841:C
11837:F
11835:*
11833:M
11831:=
11829:f
11813:.
11810:)
11804:2
11801:(
11792:F
11786:=
11783:)
11777:(
11767:)
11764:F
11755:M
11751:(
11727:.
11714:M
11711:=
11703:2
11663:=
11660:)
11657:0
11654:(
11646:2
11632:M
11628:=
11625:)
11619:(
11616:b
11586:)
11580:(
11577:b
11574:=
11566:2
11552:M
11527:.
11518:M
11512:2
11504:4
11501:=
11496:3
11482:M
11459:.
11456:V
11453:d
11449:)
11446:F
11440:M
11437:(
11431:f
11424:2
11420:H
11411:=
11408:V
11405:d
11401:F
11395:)
11392:f
11389:M
11386:(
11379:3
11375:H
11349:,
11346:)
11343:F
11334:M
11330:(
11324:f
11321:=
11318:)
11315:F
11309:)
11306:f
11303:M
11300:(
11297:(
11288:M
11277:f
11263:A
11260:d
11256:F
11247:M
11239:2
11235:H
11226:=
11223:V
11220:d
11216:F
11209:3
11205:H
11177:.
11174:)
11171:2
11167:/
11163:t
11160:(
11157:F
11154:Q
11151:=
11148:)
11145:t
11142:(
11139:F
11130:M
11119:F
11117:*
11115:M
11111:t
11107:r
11099:F
11095:F
11087:1
11084:K
11068:.
11065:A
11062:d
11058:F
11055:Q
11050:2
11046:/
11042:1
11038:)
11032:2
11028:x
11024:+
11019:2
11015:b
11011:+
11008:1
11005:(
10998:2
10994:H
10985:=
10982:V
10979:d
10975:F
10968:3
10964:H
10948:y
10934:s
10931:d
10925:s
10921:g
10914:F
10907:1
10903:K
10894:=
10891:F
10888:Q
10874:H
10870:F
10865:s
10861:g
10857:y
10846:f
10843:0
10840:M
10832:M
10828:0
10825:M
10809:.
10806:f
10801:2
10791:0
10787:M
10783:4
10780:=
10777:f
10772:0
10768:M
10762:3
10743:0
10740:M
10722:,
10719:f
10714:2
10706:S
10703:4
10700:=
10697:f
10694:S
10691:)
10686:2
10682:R
10673:2
10665:(
10655:t
10651:f
10647:t
10645:(
10627:.
10622:t
10614:)
10611:t
10608:2
10605:(
10596:2
10593:+
10588:2
10583:t
10575:=
10570:t
10562:)
10559:t
10550:+
10547:t
10538:(
10535:+
10530:2
10525:t
10498:.
10495:f
10490:t
10482:)
10479:t
10470:+
10467:t
10458:(
10455:+
10452:f
10447:2
10442:t
10434:=
10431:f
10426:r
10418:r
10415:+
10412:f
10407:t
10399:t
10390:+
10387:f
10382:2
10377:t
10369:=
10366:f
10363:)
10358:2
10354:R
10350:+
10345:2
10334:(
10322:t
10318:r
10314:f
10310:f
10293:.
10290:f
10287:)
10282:2
10278:R
10269:2
10261:(
10258:E
10255:=
10252:f
10249:E
10244:3
10214:.
10211:)
10208:x
10205:,
10202:b
10199:(
10196:f
10193:=
10190:)
10187:y
10184:,
10181:x
10178:,
10175:b
10172:(
10169:f
10166:E
10150:H
10146:x
10144:,
10142:b
10140:(
10138:f
10122:.
10119:t
10116:d
10112:t
10103:)
10100:t
10097:(
10094:f
10070:2
10067:=
10064:V
10061:d
10057:f
10050:2
10046:H
10034:,
10031:t
10028:d
10024:t
10016:2
10008:)
10005:t
10002:(
9999:F
9975:4
9972:=
9969:V
9966:d
9962:F
9955:3
9951:H
9933:t
9929:r
9925:H
9921:H
9917:F
9900:.
9895:y
9887:y
9884:+
9879:x
9871:x
9868:+
9863:b
9855:b
9852:=
9847:3
9843:R
9836:,
9831:2
9826:y
9818:+
9813:2
9808:x
9800:+
9795:2
9790:b
9782:=
9777:3
9773:L
9746:x
9738:x
9735:+
9730:b
9722:b
9719:=
9714:2
9710:R
9703:,
9698:2
9693:x
9685:+
9680:2
9675:b
9667:=
9662:2
9658:L
9632:,
9627:n
9623:R
9619:)
9616:1
9610:n
9607:(
9599:2
9594:n
9590:R
9581:n
9577:L
9570:=
9565:n
9531:t
9522:=
9519:r
9509:r
9505:x
9501:b
9497:y
9493:x
9489:b
9485:r
9469:,
9466:x
9463:d
9459:b
9456:d
9450:2
9446:/
9442:1
9435:)
9429:2
9425:r
9421:+
9418:1
9415:(
9412:=
9409:A
9406:d
9400:,
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9300:,
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8521:(
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8469:,
8463:(
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8422:2
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8218:(
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8104:(
8099:0
8070:.
8067:)
8064:)
8061:x
8058:(
8055:g
8052:(
8041:i
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8022:d
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7985:1
7978:g
7974:(
7940:2
7934:H
7921:R
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7913:λ
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7880:=
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7867:F
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7849:(
7839:C
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7600:{
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7554:.
7540:)
7533:4
7530:1
7524:+
7519:2
7510:(
7506:=
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7448:d
7443:)
7438:2
7425:(
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7388:2
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7310:1
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7264:i
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7147:(
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7132:)
7129:x
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7095:=
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7089:x
7086:(
7083:f
7071:F
7067:C
7063:F
7061:*
7059:1
7056:M
7054:=
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7036:.
7033:)
7027:(
7018:F
7012:=
7009:)
7003:(
6993:)
6990:F
6980:1
6976:M
6972:(
6946:,
6943:A
6940:d
6923:f
6917:=
6914:)
6908:(
6899:f
6872:2
6866:H
6841:.
6826:1
6822:M
6818:=
6784:.
6778:d
6769:i
6763:1
6755:)
6742:t
6730:t
6720:(
6710:2
6705:0
6691:2
6687:1
6682:=
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6676:i
6671:t
6667:e
6663:(
6641:r
6629:i
6627:(
6625:λ
6597:1
6593:M
6587:1
6580:)
6573:(
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6567:=
6541:R
6537:R
6523:t
6513:/
6507:t
6501:i
6497:e
6474:,
6469:2
6440:=
6437:t
6434:d
6427:t
6413:t
6395:=
6392:)
6389:i
6386:(
6377:f
6373:=
6370:)
6364:(
6361:b
6339:.
6330:f
6325:)
6318:4
6315:1
6309:+
6304:2
6295:(
6291:=
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6276:2
6234:1
6230:M
6226:=
6217:f
6194:.
6184:i
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6175:)
6168:4
6165:3
6159:+
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6141:=
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6106:f
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6088:A
6085:d
6081:)
6078:F
6068:0
6064:M
6060:(
6054:f
6047:2
6041:H
6030:=
6027:V
6024:d
6020:F
6014:)
6011:f
6006:0
6002:M
5998:(
5991:3
5985:H
5967:f
5965:*
5963:M
5959:0
5956:M
5940:.
5937:A
5934:d
5930:)
5927:F
5917:1
5913:M
5909:(
5903:f
5896:2
5890:H
5879:=
5876:V
5873:d
5869:F
5863:)
5860:f
5855:1
5851:M
5847:(
5840:3
5834:H
5802:y
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5795:)
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5789:,
5786:y
5783:,
5780:x
5777:(
5774:F
5751:2
5747:/
5743:1
5739:r
5735:=
5732:)
5729:r
5726:,
5723:x
5720:(
5717:F
5707:1
5703:M
5690:1
5687:M
5671:.
5667:)
5660:4
5657:3
5651:+
5646:2
5637:(
5631:0
5627:M
5623:=
5618:0
5614:M
5608:3
5591:f
5588:1
5585:M
5581:0
5578:M
5574:3
5570:C
5551:n
5545:H
5531:n
5514:,
5511:f
5507:)
5500:4
5497:3
5491:+
5486:2
5477:(
5471:1
5467:M
5463:=
5460:f
5455:1
5451:M
5445:3
5416:)
5413:r
5410:,
5407:x
5404:(
5401:f
5393:2
5389:/
5385:1
5381:r
5377:=
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5371:r
5368:,
5365:y
5362:,
5359:x
5356:(
5353:f
5348:1
5344:M
5319:2
5313:H
5300:r
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5238:.
5235:t
5232:d
5228:|
5224:t
5214:|
5210:)
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5204:(
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5180:=
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5174:d
5170:f
5145:.
5140:t
5132:t
5118:2
5113:t
5102:=
5089:t
5085:t
5081:k
5077:i
5074:e
5070:k
5054:2
5048:H
5023:.
5020:)
5015:2
5010:r
5002:+
4997:2
4992:x
4984:(
4979:2
4975:r
4968:=
4938:r
4935:d
4931:x
4928:d
4922:2
4915:r
4911:=
4908:A
4905:d
4880:)
4874:2
4870:r
4866:d
4863:+
4858:2
4854:x
4850:d
4846:(
4840:2
4833:r
4829:=
4824:2
4820:s
4816:d
4801:G
4797:K
4793:G
4777:2
4771:H
4758:K
4754:i
4738:.
4733:1
4726:)
4722:d
4719:+
4716:w
4713:c
4710:(
4707:)
4704:b
4701:+
4698:w
4695:a
4692:(
4689:=
4686:)
4683:w
4680:(
4677:g
4651:)
4645:d
4640:c
4633:b
4628:a
4622:(
4617:=
4614:g
4586:}
4583:0
4577:r
4571:i
4568:r
4565:+
4562:x
4559:{
4556:=
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4545:H
4530:R
4526:G
4500:)
4495:3
4489:H
4483:(
4478:2
4474:L
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4441:z
4438:(
4433:0
4425:)
4422:g
4419:(
4406:)
4403:g
4400:(
4397:f
4392:K
4388:/
4384:G
4376:=
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4370:z
4367:,
4361:(
4358:f
4355:W
4343:d
4339:)
4337:C
4333:L
4318:)
4313:3
4307:H
4301:(
4296:2
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4281:W
4277:λ
4261:.
4258:)
4253:0
4245:,
4240:0
4232:)
4229:g
4226:(
4213:(
4210:=
4207:)
4204:g
4201:(
4165:2
4157:)
4151:2
4146:|
4141:z
4137:|
4133:+
4130:1
4126:(
4119:1
4108:=
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4102:z
4099:(
4094:0
4065:.
4062:)
4059:)
4056:z
4053:(
4050:g
4047:(
4036:i
4030:2
4022:|
4017:d
4014:+
4011:z
4008:c
4004:|
4000:=
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3994:z
3991:(
3985:)
3980:1
3973:g
3969:(
3935:3
3929:H
3916:C
3912:C
3908:λ
3904:G
3888:.
3882:|
3868:|
3864:R
3860:e
3856:C
3849:|
3845:)
3839:(
3836:F
3832:|
3819:C
3811:R
3787:.
3779:}
3768:|
3764:)
3761:t
3758:(
3749:)
3746:t
3743:(
3740:f
3735:M
3731:)
3724:+
3721:I
3718:(
3713:N
3709:)
3703:2
3699:t
3695:+
3692:1
3689:(
3685:|
3679:t
3670:|
3666:f
3662:{
3658:=
3653:S
3639:f
3635:R
3616:.
3603:)
3600:1
3597:+
3592:2
3584:(
3581:=
3552:λ
3529:f
3506:)
3503:K
3499:/
3495:G
3489:K
3486:(
3481:1
3477:L
3470:f
3435:f
3426:f
3423::
3420:U
3413:)
3407:d
3401:2
3393:,
3389:R
3385:(
3380:2
3376:L
3369:)
3366:K
3362:/
3358:G
3352:K
3349:(
3344:2
3340:L
3336::
3333:U
3327:{
3297:.
3291:d
3285:2
3270:)
3264:(
3259:2
3249:f
3239:)
3233:(
3228:1
3218:f
3208:=
3205:g
3202:d
3191:2
3187:f
3179:1
3175:f
3169:G
3149:i
3129:)
3124:1
3117:g
3113:(
3110:f
3104:=
3101:)
3098:g
3095:(
3086:f
3075:K
3071:G
3069:(
3067:c
3062:i
3058:f
3042:1
3038:f
3024:2
3020:f
3016:=
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2991:.
2985:d
2979:2
2971:)
2968:x
2965:(
2952:)
2946:(
2937:f
2928:=
2925:)
2922:x
2919:(
2916:f
2892:V
2889:d
2872:f
2866:=
2863:)
2857:(
2848:f
2821:,
2815:t
2801:t
2786:=
2783:)
2780:t
2777:(
2755:f
2751:R
2724:+
2716:2
2712:t
2708:d
2702:2
2698:d
2689:=
2686:U
2674:U
2661:t
2657:t
2655:(
2653:f
2649:t
2647:(
2641:R
2625:.
2622:t
2619:d
2615:t
2607:2
2598:)
2595:t
2592:(
2589:f
2568:=
2565:V
2562:d
2558:f
2533:.
2528:t
2520:t
2511:2
2503:2
2498:t
2487:=
2474:t
2470:t
2466:k
2462:j
2459:e
2457:(
2455:k
2439:3
2433:H
2408:.
2403:r
2395:r
2392:+
2389:)
2384:2
2379:r
2371:+
2366:2
2361:y
2353:+
2348:2
2343:x
2335:(
2330:2
2326:r
2319:=
2289:r
2286:d
2282:y
2279:d
2275:x
2272:d
2266:3
2259:r
2255:=
2252:V
2249:d
2224:)
2218:2
2214:r
2210:d
2207:+
2202:2
2198:y
2194:d
2191:+
2186:2
2182:x
2178:d
2174:(
2168:2
2161:r
2157:=
2152:2
2148:s
2144:d
2129:G
2115:.
2112:K
2108:/
2104:G
2101:=
2096:3
2090:H
2077:K
2073:j
2057:.
2052:1
2045:)
2041:d
2038:+
2035:w
2032:c
2029:(
2026:)
2023:b
2020:+
2017:w
2014:a
2011:(
2008:=
2005:)
2002:w
1999:(
1996:g
1974:)
1968:d
1963:c
1956:b
1951:a
1945:(
1940:=
1937:g
1913:}
1910:0
1904:t
1898:j
1895:t
1892:+
1889:i
1886:y
1883:+
1880:x
1877:{
1874:=
1869:3
1863:H
1846:C
1842:G
1830:A
1826:μ
1822:λ
1802:.
1799:)
1796:1
1793:,
1790:1
1787:)
1784:g
1781:(
1775:(
1772:=
1769:)
1766:g
1763:(
1743:M
1739:K
1735:f
1719:.
1710:k
1707:d
1701:2
1696:|
1691:)
1688:k
1685:(
1682:f
1678:|
1672:K
1664:=
1659:2
1651:f
1628:,
1625:)
1622:x
1617:1
1610:g
1606:(
1603:f
1600:=
1597:)
1594:x
1591:(
1588:f
1585:)
1582:g
1579:(
1566:B
1562:b
1548:)
1545:g
1542:(
1539:f
1536:)
1533:b
1530:(
1522:2
1518:/
1514:1
1510:)
1506:b
1503:(
1497:=
1494:)
1491:b
1488:g
1485:(
1482:f
1472:G
1468:f
1464:A
1452:B
1445:G
1441:λ
1437:K
1433:A
1425:M
1421:G
1413:λ
1396:.
1393:W
1389:A
1385:X
1375:.
1365:K
1361:A
1343:,
1340:)
1337:A
1334:(
1329:K
1325:C
1320:/
1316:)
1313:A
1310:(
1305:K
1301:N
1297:=
1294:W
1285:A
1277:s
1273:s
1271:·
1269:2
1265:1
1261:2
1257:1
1252:.
1232:x
1228:K
1224:k
1210:)
1207:x
1204:(
1196:2
1192:/
1188:1
1184:)
1180:x
1177:(
1172:N
1169:A
1161:=
1158:)
1155:x
1152:k
1149:(
1131:A
1116:G
1111:;
1109:G
1103:=
1101:G
1094:A
1090:T
1088:,
1086:A
1082:A
1077:;
1075:K
1050:.
1047:k
1044:d
1038:1
1031:)
1027:k
1024:g
1021:(
1006:K
998:=
995:)
992:g
989:(
958:G
954:λ
947:G
927:A
915:f
911:K
909:/
907:G
905:\
903:K
901:(
899:c
896:C
892:f
878:.
875:g
872:d
868:)
865:g
862:(
846:)
843:g
840:(
837:f
832:G
824:=
821:)
818:)
815:f
812:(
806:(
786:G
782:λ
780:φ
771:X
755:A
743:λ
736:G
732:K
728:K
726:/
724:G
722:\
720:K
718:(
716:L
712:0
709:H
691:U
668:A
656:X
654:(
652:L
648:0
645:H
641:U
637:X
621:0
618:H
610:H
606:K
602:0
599:H
583:A
571:H
567:G
541:A
528:H
524:S
497:A
481:C
465:A
453:X
430:A
407:K
403:G
399:K
395:G
393:(
391:L
389:=
387:H
380:G
376:K
372:K
370:/
368:G
366:\
364:K
362:(
360:c
357:C
350:G
342:K
331:G
313:R
301:K
299:/
297:G
272:"
262:K
260:/
258:G
250:G
246:d
242:c
238:λ
234:c
207:λ
199:G
195:K
193:/
191:G
172:K
165:K
163:/
161:G
145:G
100:)
98:X
96:(
94:L
77:X
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