22:
2891:
393:
characterization. In fact, the matrix coefficients form a unital algebra invariant under complex conjugation because the product of two matrix coefficients is a matrix coefficient of the tensor product representation, and the complex conjugate is a matrix coefficient of the dual representation.
2713:
1939:
997:
decomposes into an orthogonal direct sum of all the irreducible unitary representations, in which the multiplicity of each irreducible representation is equal to its degree (that is, the dimension of the underlying space of the representation). Thus,
1094:
1614:
2886:{\displaystyle \operatorname {SU} (2)=\left\{{\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}}:\ \ \alpha ,\beta \in \mathbb {C} ,\,|\alpha |^{2}+|\beta |^{2}=1\right\}~,}
652:
3274:
1798:
3339:
2392:
2295:
445:
into finite-dimensional representations. Now, intuitively groups were conceived as rotations on geometric objects, so it is only natural to study representations which essentially arise from continuous
3419:
2530:
3510:
2604:
1451:
2965:
1698:
1367:
1275:
1821:
896:
339:) as an open subset. Matrix coefficients are continuous, since representations are by definition continuous, and linear functionals on finite-dimensional spaces are also continuous.
294:
1004:
1524:
2076:
1216:
3421:
consisting of matrix coefficients amounts to finding an orthonormal basis consisting of hyperspherical harmonics, which is a standard construction in analysis on spheres.
2650:
2206:
1655:
as follows. Suppose that a representative π is chosen for each isomorphism class of irreducible unitary representation, and denote the collection of all such π by Σ. Let
1981:
1516:
1328:
1177:
2693:
2156:
1653:
1490:
1302:
1151:
995:
943:
817:
764:
255:
3120:
2459:
3140:
3194:
3167:
3080:
2921:
2226:
2176:
203:) decomposes as the direct sum of all irreducible unitary representations. Moreover, the matrix coefficients of the irreducible unitary representations form an
3530:
3100:
3053:
3029:
3009:
2985:
2116:
2096:
2008:
1236:
2695:
in terms of matrix coefficients as a generalization of the theory of
Fourier series. Indeed, this decomposition is often referred to as a Fourier series.
366:
456:, this approach is similar except that the circle group is (ultimately) generalised to the group of unitary operators on a given Hilbert space.)
452:
on
Hilbert spaces. (For those who were first introduced to dual groups consisting of characters which are the continuous homomorphisms into the
583:
3202:
1706:
3282:
2306:
3938:
2234:
394:
Hence the theorem follows directly from the Stone–Weierstrass theorem if the matrix coefficients separate points, which is obvious if
3439:
2417:
3347:
3933:
3923:
3472:: Every compact Lie group has a faithful finite-dimensional representation and is therefore isomorphic to a closed subgroup of
3669:
Peter, F.; Weyl, H. (1927), "Die Vollständigkeit der primitiven
Darstellungen einer geschlossenen kontinuierlichen Gruppe",
2467:
3475:
2541:
1375:
3718:
2926:
3943:
3780:
3758:
3699:
2609:
The last part of the Peter–Weyl theorem then asserts in this case that these functions form an orthonormal basis for
2158:, and therefore a Hilbert space in its own right. Within the space of matrix coefficients for a fixed representation
1934:{\displaystyle \left\{{\sqrt {d^{(\pi )}}}u_{ij}^{(\pi )}\mid \,\pi \in \Sigma ,\,\,1\leq i,j\leq d^{(\pi )}\right\}}
65:
43:
1658:
1333:
1241:
389:
in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an
36:
3828:
829:
3738:
2704:
705:
386:
1089:{\displaystyle L^{2}(G)={\underset {\pi \in \Sigma }{\widehat {\bigoplus }}}E_{\pi }^{\oplus \dim E_{\pi }}}
124:). The theorem is a collection of results generalizing the significant facts about the decomposition of the
3032:
1179:, with the two factors acting by translation on the left and the right, respectively. Fix a representation
1609:{\displaystyle L^{2}(G)={\underset {\pi \in \Sigma }{\widehat {\bigoplus }}}E_{\pi }\otimes E_{\pi }^{*}.}
267:
3928:
3733:
3728:
219:
is the group of unit complex numbers, this last result is simply a standard result from
Fourier series.
147:
be a compact group. The theorem has three parts. The first part states that the matrix coefficients of
3569:
820:
726:
448:
180:
148:
3442:
of the irreducible representations of a connected compact Lie group. The argument also depends on the
2021:
3454:
1182:
133:
3443:
503:
30:
2612:
2184:
3608:
3593:
2988:
2179:
164:
3447:
1947:
1495:
1307:
1156:
780:
438:
184:
125:
47:
2662:
2125:
1622:
1459:
1280:
1120:
964:
912:
786:
733:
674:). Notice how this generalises the special case of the one-dimensional Hilbert space, where U(
240:
3795:
3105:
2967:. The irreducible representations of SU(2), meanwhile, are labeled by a non-negative integer
2431:
3125:
3172:
3145:
3058:
2899:
2652:. In this case, the theorem is simply a standard result from the theory of Fourier series.
1099:
where Σ denotes the set of (isomorphism classes of) irreducible unitary representations of
312:
3434:
The Peter–Weyl theorem—specifically the assertion that the characters form an orthonormal
2211:
2161:
8:
3541:
From the Peter–Weyl theorem, one can deduce a significant general structure theorem. Let
2428:
A simple but helpful example is the case of the group of complex numbers of magnitude 1,
1815:
be the degree of the representation π. The theorem now asserts that the set of functions
549:
308:
721:
To state the third and final part of the theorem, there is a natural
Hilbert space over
3845:
3812:
3656:
3515:
3085:
3038:
3014:
2994:
2970:
2101:
2081:
1993:
1221:
1104:
534:. Conversely, given such a map, we can uniquely recover the action in the obvious way.
229:
3776:
3754:
3714:
3695:
2300:
In the notation above, the character is the sum of the diagonal matrix coefficients:
950:
906:
324:
204:
114:
94:
90:
3837:
3826:; Stewart, T. E. (1961), "The cohomology of differentiable transformation groups",
3804:
3768:
3746:
3678:
569:
3644:
3612:
3546:
3142:. The key to verifying this claim is to compute that for any two complex numbers
2461:. In this case, the irreducible representations are one-dimensional and given by
1238:. The space of matrix coefficients for the representation may be identified with
681:
Given these definitions, we can state the second part of the Peter–Weyl theorem (
2535:
There is then a single matrix coefficient for each representation, the function
3823:
406:, p. 17). Conversely, it is a consequence of the theorem that any compact
3597:
3917:
3790:
3775:, Progress in Mathematics, vol. 140 (2nd ed.), Boston: Birkhäuser,
102:
98:
2122:. The space of square-integrable class functions forms a closed subspace of
3438:
for the space of square-integrable class functions—plays a key role in the
2408:
form a
Hilbert basis for the space of square-integrable class functions on
768:
453:
437:
The second part of the theorem gives the existence of a decomposition of a
399:
378:
328:
129:
106:
3713:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
3711:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
137:
110:
82:
3429:
1700:
be the matrix coefficients of π in an orthonormal basis, in other words
3849:
3816:
3682:
647:{\displaystyle \langle \rho (g)v,\rho (g)w\rangle =\langle v,w\rangle }
3643:. It may of course not itself be a Lie group: it may for example be a
3465:
One important consequence of the Peter–Weyl theorem is the following:
3269:{\displaystyle (\alpha ,\beta )\mapsto (z_{1}\alpha +z_{2}\beta )^{m}}
1793:{\displaystyle u_{ij}^{(\pi )}(g)=\langle \pi (g)e_{j},e_{i}\rangle .}
3585:
3334:{\displaystyle (\alpha ,\beta )\in \mathbb {C} ^{2}=\mathbb {R} ^{4}}
2387:{\displaystyle \chi _{\pi }=\sum _{i=1}^{d^{(\pi )}}u_{ii}^{(\pi )}.}
407:
362:
3841:
3808:
2416:
This result plays an important part in Weyl's classification of the
2987:
and can be realized as the natural action of SU(2) on the space of
2397:
An important consequence of the preceding result is the following:
730:
191:. The third part then asserts that the regular representation of
2290:{\displaystyle \chi _{\pi }(g)=\operatorname {trace} (\pi (g)).}
716:
78:
Basic result in harmonic analysis on compact topological groups
3793:(1961), "Cohomology of topological groups and solvmanifolds",
417:
A corollary of this result is that the matrix coefficients of
708:
of irreducible finite-dimensional unitary representations of
432:
3536:
3414:{\displaystyle L^{2}(\operatorname {SU} (2))=L^{2}(S^{3})}
3011:
in two complex variables. The matrix coefficients of the
1330:
on the matrix coefficients corresponds to the action on
183:. The second part asserts the complete reducibility of
2404:: The characters of the irreducible representations of
945:. Roughly it asserts that the matrix coefficients for
2745:
1662:
3518:
3478:
3430:
Representation theory of connected compact Lie groups
3350:
3285:
3205:
3175:
3148:
3128:
3108:
3088:
3061:
3041:
3017:
2997:
2973:
2929:
2902:
2716:
2665:
2615:
2544:
2525:{\displaystyle \pi _{n}(e^{i\theta })=e^{in\theta }.}
2470:
2434:
2309:
2237:
2214:
2187:
2164:
2128:
2104:
2084:
2024:
1996:
1950:
1824:
1709:
1661:
1625:
1527:
1498:
1462:
1378:
1336:
1310:
1283:
1244:
1224:
1185:
1159:
1123:
1007:
967:
915:
832:
789:
736:
692:
Let ρ be a unitary representation of a compact group
586:
270:
243:
3505:{\displaystyle \operatorname {GL} (n;\mathbb {C} )}
3836:(4), The Johns Hopkins University Press: 623–644,
3524:
3504:
3460:
3413:
3333:
3268:
3188:
3161:
3134:
3114:
3094:
3074:
3047:
3023:
3003:
2979:
2959:
2915:
2885:
2687:
2644:
2599:{\displaystyle u_{n}(e^{i\theta })=e^{in\theta }.}
2598:
2524:
2453:
2386:
2289:
2220:
2200:
2170:
2150:
2110:
2090:
2070:
2002:
1975:
1933:
1792:
1692:
1647:
1608:
1510:
1484:
1446:{\displaystyle (g,h)\cdot A=\pi (g)A\pi (h)^{-1}.}
1445:
1361:
1322:
1296:
1269:
1230:
1210:
1171:
1145:
1088:
989:
937:
890:
811:
758:
646:
342:The first part of the Peter–Weyl theorem asserts (
288:
249:
2960:{\displaystyle \mathbb {C} ^{2}=\mathbb {R} ^{4}}
1985:
3915:
3545:be a compact topological group, which we assume
2703:We use the standard representation of the group
2418:representations of a connected compact Lie group
1304:to itself. The natural left and right action of
1153:as a representation of the direct product group
3344:In this case, finding an orthonormal basis for
901:The final statement of the Peter–Weyl theorem (
3082:of homogeneous harmonic polynomials of degree
1693:{\displaystyle \scriptstyle {u_{ij}^{(\pi )}}}
1619:Finally, we may form an orthonormal basis for
1362:{\displaystyle \operatorname {End} (E_{\pi })}
1270:{\displaystyle \operatorname {End} (E_{\pi })}
3822:
1784:
1746:
717:Decomposition of square-integrable functions
641:
629:
623:
587:
552:, ρ, which arise from continuous actions of
2896:Thus, SU(2) is represented as the 3-sphere
3751:Representation theory of semisimple groups
522:. This map is clearly a homomorphism from
3668:
3495:
3321:
3306:
2947:
2932:
2822:
2815:
1891:
1890:
1877:
433:Decomposition of a unitary representation
121:
66:Learn how and when to remove this message
3803:(1), Princeton University Press: 20–48,
3453:An outline of the argument may be found
891:{\displaystyle \rho (h)f(g)=f(h^{-1}g).}
29:This article includes a list of general
3537:Structure of compact topological groups
3916:
3789:
3572:on the left, we consider the image of
1107:of the direct sum of the total spaces
222:
3905:
3767:
3745:
2659:, we can regard the decomposition of
902:
682:
411:
403:
347:
3893:
3881:
3869:
3708:
3689:
2698:
560:. We say that a representation ρ is
470:A continuous linear action ∗ :
343:
289:{\displaystyle \varphi =L\circ \pi }
15:
2423:
1554:
1034:
530:), the bounded linear operators on
89:is a basic result in the theory of
13:
3584:is compact, and a subgroup of the
1884:
1569:
1049:
905:, Theorem 1.12) gives an explicit
666:. (I.e. it is unitary if ρ :
482:, gives rise to a continuous map ρ
357:The set of matrix coefficients of
35:it lacks sufficient corresponding
14:
3955:
3939:Theorems in representation theory
3773:Lie Groups Beyond an Introduction
410:is isomorphic to a matrix group (
2071:{\displaystyle f(hgh^{-1})=f(g)}
1103:, and the summation denotes the
949:, suitably renormalized, are an
385:This first result resembles the
335:(e.g. trace), which contains GL(
20:
3829:American Journal of Mathematics
3461:Linearity of compact Lie groups
3424:
3055:, that is, the restrictions to
1211:{\displaystyle (\pi ,E_{\pi })}
767:; this makes sense because the
3899:
3887:
3875:
3863:
3753:, Princeton University Press,
3499:
3485:
3446:(for class functions) and the
3408:
3395:
3379:
3376:
3370:
3361:
3298:
3286:
3257:
3224:
3221:
3218:
3206:
2856:
2847:
2833:
2824:
2729:
2723:
2682:
2676:
2639:
2626:
2571:
2555:
2497:
2481:
2376:
2370:
2350:
2344:
2281:
2278:
2272:
2266:
2254:
2248:
2145:
2139:
2065:
2059:
2050:
2028:
1986:Restriction to class functions
1967:
1961:
1921:
1915:
1869:
1863:
1843:
1837:
1758:
1752:
1740:
1734:
1729:
1723:
1683:
1677:
1642:
1636:
1544:
1538:
1479:
1473:
1428:
1421:
1412:
1406:
1391:
1379:
1356:
1343:
1277:, the space of linear maps of
1264:
1251:
1205:
1186:
1140:
1134:
1024:
1018:
984:
978:
932:
926:
882:
863:
854:
848:
842:
836:
806:
800:
753:
747:
617:
611:
599:
593:
113:, in the setting of a compact
1:
3934:Theorems in harmonic analysis
3924:Unitary representation theory
3662:
3549:. For any finite-dimensional
3279:is harmonic as a function of
1492:as unitary representation of
690:Peter–Weyl Theorem (Part II).
678:) is just the circle group.)
237:is a complex-valued function
171:, and thus also in the space
105:. It was initially proved by
2780:
2761:
2645:{\displaystyle L^{2}(S^{1})}
2201:{\displaystyle \chi _{\pi }}
367:continuous complex functions
355:Peter–Weyl Theorem (Part I).
7:
3734:Encyclopedia of Mathematics
3650:
3641:inverse limit of Lie groups
1944:is an orthonormal basis of
727:square-integrable functions
696:on a complex Hilbert space
463:be a topological group and
307:) is a finite-dimensional (
181:square-integrable functions
149:irreducible representations
10:
3960:
1114:of the representations π.
704:splits into an orthogonal
101:, but are not necessarily
1976:{\displaystyle L^{2}(G).}
1511:{\displaystyle G\times G}
1323:{\displaystyle G\times G}
1172:{\displaystyle G\times G}
467:a complex Hilbert space.
387:Stone–Weierstrass theorem
261:given as the composition
134:Ferdinand Georg Frobenius
3944:Theorems in group theory
3619:, we get a result about
3033:hyperspherical harmonics
2688:{\displaystyle L^{2}(G)}
2151:{\displaystyle L^{2}(G)}
1648:{\displaystyle L^{2}(G)}
1485:{\displaystyle L^{2}(G)}
1297:{\displaystyle E_{\pi }}
1146:{\displaystyle L^{2}(G)}
990:{\displaystyle L^{2}(G)}
938:{\displaystyle L^{2}(G)}
812:{\displaystyle L^{2}(G)}
759:{\displaystyle L^{2}(G)}
250:{\displaystyle \varphi }
165:complex-valued functions
3709:Hall, Brian C. (2015),
3615:) over all such spaces
3580:). It is closed, since
3115:{\displaystyle \alpha }
2989:homogeneous polynomials
2454:{\displaystyle G=S^{1}}
327:on the vector space of
185:unitary representations
155:are dense in the space
50:more precise citations.
3526:
3506:
3448:Weyl character formula
3415:
3335:
3270:
3190:
3163:
3136:
3135:{\displaystyle \beta }
3116:
3096:
3076:
3049:
3031:th representation are
3025:
3005:
2981:
2961:
2917:
2887:
2689:
2655:For any compact group
2646:
2600:
2526:
2455:
2388:
2356:
2291:
2222:
2202:
2172:
2152:
2112:
2092:
2072:
2004:
1977:
1935:
1794:
1694:
1649:
1610:
1512:
1486:
1456:Then we may decompose
1447:
1363:
1324:
1298:
1271:
1232:
1212:
1173:
1147:
1090:
991:
939:
892:
813:
781:unitary representation
760:
714:
648:
439:unitary representation
383:
290:
251:
126:regular representation
3796:Annals of Mathematics
3690:Bump, Daniel (2004),
3604:is a Lie group also.
3527:
3507:
3444:Weyl integral formula
3416:
3336:
3271:
3191:
3189:{\displaystyle z_{2}}
3164:
3162:{\displaystyle z_{1}}
3137:
3117:
3097:
3077:
3075:{\displaystyle S^{3}}
3050:
3026:
3006:
2982:
2962:
2918:
2916:{\displaystyle S^{3}}
2888:
2690:
2647:
2601:
2527:
2456:
2389:
2323:
2292:
2223:
2203:
2173:
2153:
2113:
2093:
2073:
2005:
1978:
1936:
1795:
1695:
1650:
1611:
1513:
1487:
1448:
1364:
1325:
1299:
1272:
1233:
1213:
1174:
1148:
1091:
992:
940:
893:
814:
761:
687:
649:
352:
291:
252:
122:Peter & Weyl 1927
3729:"Peter-Weyl theorem"
3553:-invariant subspace
3516:
3476:
3348:
3283:
3203:
3173:
3146:
3126:
3106:
3086:
3059:
3039:
3015:
2995:
2971:
2927:
2900:
2714:
2663:
2613:
2542:
2468:
2432:
2307:
2235:
2221:{\displaystyle \pi }
2212:
2185:
2171:{\displaystyle \pi }
2162:
2126:
2102:
2082:
2022:
1994:
1948:
1822:
1707:
1659:
1623:
1525:
1496:
1460:
1376:
1334:
1308:
1281:
1242:
1222:
1183:
1157:
1121:
1005:
965:
913:
830:
787:
734:
584:
377:, equipped with the
313:group representation
299:where π :
268:
241:
215:). In the case that
3908:, Corollary IV.4.22
3627:acts faithfully on
3607:If we now take the
3592:). It follows by a
2380:
1873:
1733:
1687:
1602:
1117:We may also regard
1085:
961:). In particular,
550:group homomorphisms
543:on a Hilbert space
539:representations of
537:Thus we define the
223:Matrix coefficients
132:, as discovered by
109:, with his student
3929:Topological groups
3824:Palais, Richard S.
3683:10.1007/BF01447892
3657:Pontryagin duality
3600:that the image of
3522:
3502:
3411:
3331:
3266:
3186:
3159:
3132:
3112:
3092:
3072:
3045:
3021:
3001:
2977:
2957:
2913:
2883:
2787:
2685:
2642:
2596:
2522:
2451:
2384:
2357:
2287:
2218:
2198:
2168:
2148:
2108:
2088:
2068:
2000:
1973:
1931:
1850:
1790:
1710:
1690:
1689:
1664:
1645:
1606:
1588:
1573:
1508:
1482:
1443:
1359:
1320:
1294:
1267:
1228:
1208:
1169:
1143:
1086:
1055:
1053:
987:
935:
888:
809:
756:
644:
286:
247:
230:matrix coefficient
95:topological groups
87:Peter–Weyl theorem
3791:Mostow, George D.
3769:Knapp, Anthony W.
3611:(in the sense of
3525:{\displaystyle n}
3095:{\displaystyle m}
3048:{\displaystyle m}
3024:{\displaystyle m}
3004:{\displaystyle m}
2980:{\displaystyle m}
2879:
2801:
2798:
2783:
2764:
2699:An example: SU(2)
2111:{\displaystyle h}
2091:{\displaystyle g}
2003:{\displaystyle f}
1848:
1560:
1551:
1231:{\displaystyle G}
1040:
1031:
951:orthonormal basis
907:orthonormal basis
823:on the left, via
685:, Theorem 1.12):
414:, Theorem 1.15).
350:, Theorem 1.12):
325:linear functional
205:orthonormal basis
115:topological group
91:harmonic analysis
76:
75:
68:
3951:
3909:
3903:
3897:
3891:
3885:
3879:
3873:
3867:
3852:
3819:
3785:
3763:
3742:
3723:
3704:
3685:
3531:
3529:
3528:
3523:
3511:
3509:
3508:
3503:
3498:
3420:
3418:
3417:
3412:
3407:
3406:
3394:
3393:
3360:
3359:
3340:
3338:
3337:
3332:
3330:
3329:
3324:
3315:
3314:
3309:
3275:
3273:
3272:
3267:
3265:
3264:
3252:
3251:
3236:
3235:
3196:, the function
3195:
3193:
3192:
3187:
3185:
3184:
3168:
3166:
3165:
3160:
3158:
3157:
3141:
3139:
3138:
3133:
3121:
3119:
3118:
3113:
3101:
3099:
3098:
3093:
3081:
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3073:
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3051:
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3010:
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3002:
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2523:
2518:
2517:
2496:
2495:
2480:
2479:
2460:
2458:
2457:
2452:
2450:
2449:
2424:An example: U(1)
2393:
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2385:
2379:
2368:
2355:
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2319:
2318:
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2049:
2048:
2009:
2007:
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2001:
1982:
1980:
1979:
1974:
1960:
1959:
1940:
1938:
1937:
1932:
1930:
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1924:
1872:
1861:
1849:
1847:
1846:
1831:
1811:. Finally, let
1799:
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1796:
1791:
1783:
1782:
1770:
1769:
1732:
1721:
1699:
1697:
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1295:
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1234:
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1209:
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1203:
1178:
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1170:
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1133:
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1095:
1093:
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1052:
1041:
1033:
1017:
1016:
996:
994:
993:
988:
977:
976:
944:
942:
941:
936:
925:
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897:
895:
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889:
878:
877:
818:
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815:
810:
799:
798:
765:
763:
762:
757:
746:
745:
653:
651:
650:
645:
570:unitary operator
494:(functions from
365:in the space of
303: → GL(
295:
293:
292:
287:
256:
254:
253:
248:
163:) of continuous
71:
64:
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57:
51:
46:this article by
37:inline citations
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3645:profinite group
3613:category theory
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2923:sitting inside
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1195:
1184:
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1128:
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831:
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788:
785:
784:
741:
737:
735:
732:
731:
719:
585:
582:
581:
509:
506:) defined by: ρ
504:strong topology
485:
435:
269:
266:
265:
242:
239:
238:
225:
79:
72:
61:
55:
52:
42:Please help to
41:
25:
21:
12:
11:
5:
3957:
3947:
3946:
3941:
3936:
3931:
3926:
3911:
3910:
3898:
3886:
3874:
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3859:
3855:
3854:
3820:
3787:
3781:
3765:
3759:
3747:Knapp, Anthony
3743:
3725:
3720:978-3319134666
3719:
3706:
3700:
3687:
3664:
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3660:
3659:
3652:
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3538:
3535:
3534:
3533:
3521:
3501:
3497:
3493:
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3487:
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3481:
3462:
3459:
3440:classification
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3405:
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2750:
2747:
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2744:
2738:
2734:
2731:
2728:
2725:
2722:
2719:
2700:
2697:
2684:
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2678:
2673:
2669:
2641:
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2632:
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2607:
2606:
2595:
2590:
2587:
2584:
2580:
2576:
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2565:
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2557:
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2548:
2533:
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2521:
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2513:
2510:
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2502:
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2448:
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2422:
2414:
2413:
2395:
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2378:
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2372:
2367:
2364:
2360:
2352:
2349:
2346:
2342:
2336:
2333:
2330:
2326:
2322:
2317:
2313:
2298:
2297:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2245:
2241:
2217:
2195:
2191:
2167:
2147:
2144:
2141:
2136:
2132:
2107:
2087:
2067:
2064:
2061:
2058:
2055:
2052:
2047:
2044:
2040:
2036:
2033:
2030:
2027:
2016:class function
1999:
1987:
1984:
1972:
1969:
1966:
1963:
1958:
1954:
1942:
1941:
1929:
1923:
1920:
1917:
1913:
1909:
1906:
1903:
1900:
1897:
1894:
1889:
1886:
1883:
1880:
1876:
1871:
1868:
1865:
1860:
1857:
1853:
1845:
1842:
1839:
1835:
1828:
1801:
1800:
1789:
1786:
1781:
1777:
1773:
1768:
1764:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1731:
1728:
1725:
1720:
1717:
1713:
1685:
1682:
1679:
1674:
1671:
1667:
1644:
1641:
1638:
1633:
1629:
1617:
1616:
1605:
1600:
1595:
1591:
1587:
1582:
1578:
1571:
1568:
1565:
1559:
1556:
1549:
1546:
1543:
1540:
1535:
1531:
1507:
1504:
1501:
1481:
1478:
1475:
1470:
1466:
1454:
1453:
1442:
1437:
1434:
1430:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1358:
1353:
1349:
1345:
1342:
1339:
1319:
1316:
1313:
1291:
1287:
1266:
1261:
1257:
1253:
1250:
1247:
1227:
1207:
1202:
1198:
1194:
1191:
1188:
1168:
1165:
1162:
1142:
1139:
1136:
1131:
1127:
1111:
1097:
1096:
1081:
1077:
1073:
1070:
1067:
1062:
1058:
1051:
1048:
1045:
1039:
1036:
1029:
1026:
1023:
1020:
1015:
1011:
986:
983:
980:
975:
971:
934:
931:
928:
923:
919:
899:
898:
887:
884:
881:
876:
873:
869:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
808:
805:
802:
797:
793:
755:
752:
749:
744:
740:
725:consisting of
718:
715:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
507:
483:
434:
431:
297:
296:
285:
282:
279:
276:
273:
246:
224:
221:
93:, applying to
77:
74:
73:
28:
26:
19:
9:
6:
4:
3:
2:
3956:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3921:
3919:
3907:
3902:
3895:
3890:
3883:
3878:
3871:
3866:
3862:
3857:
3856:
3851:
3847:
3843:
3839:
3835:
3831:
3830:
3825:
3821:
3818:
3814:
3810:
3806:
3802:
3798:
3797:
3792:
3788:
3784:
3782:0-8176-4259-5
3778:
3774:
3770:
3766:
3762:
3760:0-691-09089-0
3756:
3752:
3748:
3744:
3740:
3736:
3735:
3730:
3726:
3722:
3716:
3712:
3707:
3703:
3701:0-387-21154-3
3697:
3693:
3688:
3684:
3680:
3676:
3672:
3667:
3666:
3658:
3655:
3654:
3648:
3646:
3642:
3638:
3634:
3630:
3626:
3622:
3618:
3614:
3610:
3605:
3603:
3599:
3595:
3591:
3587:
3583:
3579:
3575:
3571:
3568:
3564:
3560:
3556:
3552:
3548:
3544:
3519:
3491:
3488:
3482:
3479:
3471:
3468:
3467:
3466:
3458:
3456:
3451:
3449:
3445:
3441:
3437:
3422:
3403:
3399:
3390:
3386:
3382:
3373:
3367:
3364:
3356:
3352:
3342:
3326:
3316:
3311:
3301:
3295:
3292:
3289:
3261:
3253:
3248:
3244:
3240:
3237:
3232:
3228:
3215:
3212:
3209:
3199:
3198:
3197:
3181:
3177:
3154:
3150:
3129:
3109:
3089:
3067:
3063:
3042:
3034:
3018:
2998:
2990:
2974:
2952:
2942:
2937:
2908:
2904:
2880:
2873:
2869:
2866:
2861:
2851:
2843:
2838:
2828:
2819:
2811:
2808:
2805:
2802:
2793:
2788:
2777:
2770:
2758:
2753:
2748:
2742:
2736:
2732:
2726:
2720:
2717:
2710:
2709:
2708:
2706:
2696:
2679:
2671:
2667:
2658:
2653:
2634:
2630:
2621:
2617:
2593:
2588:
2585:
2582:
2578:
2574:
2566:
2563:
2559:
2550:
2546:
2538:
2537:
2536:
2519:
2514:
2511:
2508:
2504:
2500:
2492:
2489:
2485:
2476:
2472:
2464:
2463:
2462:
2446:
2442:
2438:
2435:
2421:
2419:
2411:
2407:
2403:
2400:
2399:
2398:
2381:
2373:
2365:
2362:
2358:
2347:
2340:
2334:
2331:
2328:
2324:
2320:
2315:
2311:
2303:
2302:
2301:
2284:
2275:
2269:
2263:
2260:
2257:
2251:
2243:
2239:
2231:
2230:
2229:
2228:, defined by
2215:
2193:
2189:
2181:
2165:
2142:
2134:
2130:
2121:
2105:
2085:
2062:
2056:
2053:
2045:
2042:
2038:
2034:
2031:
2025:
2017:
2013:
1997:
1983:
1970:
1964:
1956:
1952:
1927:
1918:
1911:
1907:
1904:
1901:
1898:
1895:
1892:
1887:
1881:
1878:
1874:
1866:
1858:
1855:
1851:
1840:
1833:
1826:
1818:
1817:
1816:
1814:
1810:
1807: ∈
1806:
1787:
1779:
1775:
1771:
1766:
1762:
1755:
1749:
1743:
1737:
1726:
1718:
1715:
1711:
1703:
1702:
1701:
1680:
1672:
1669:
1665:
1639:
1631:
1627:
1603:
1598:
1593:
1589:
1585:
1580:
1576:
1566:
1563:
1557:
1547:
1541:
1533:
1529:
1521:
1520:
1519:
1505:
1502:
1499:
1476:
1468:
1464:
1440:
1435:
1432:
1424:
1418:
1415:
1409:
1403:
1400:
1397:
1394:
1388:
1385:
1382:
1372:
1371:
1370:
1351:
1347:
1340:
1337:
1317:
1314:
1311:
1289:
1285:
1259:
1255:
1248:
1245:
1225:
1200:
1196:
1192:
1189:
1166:
1163:
1160:
1137:
1129:
1125:
1115:
1110:
1106:
1102:
1079:
1075:
1071:
1068:
1065:
1060:
1056:
1046:
1043:
1037:
1027:
1021:
1013:
1009:
1001:
1000:
999:
981:
973:
969:
960:
956:
952:
948:
929:
921:
917:
908:
904:
885:
879:
874:
871:
867:
860:
857:
851:
845:
839:
833:
826:
825:
824:
822:
803:
795:
791:
782:
778:
774:
770:
766:
750:
742:
738:
728:
724:
713:
711:
707:
703:
699:
695:
691:
686:
684:
679:
677:
673:
669:
665:
662: ∈
661:
657:
638:
635:
632:
626:
620:
614:
608:
605:
602:
596:
590:
579:
576: ∈
575:
571:
567:
563:
559:
555:
551:
547:
546:
542:
535:
533:
529:
525:
521:
517:
513:
505:
501:
497:
493:
489:
481:
477:
473:
468:
466:
462:
457:
455:
451:
450:
444:
440:
430:
428:
424:
421:are dense in
420:
415:
413:
409:
405:
401:
397:
392:
388:
382:
380:
376:
372:
368:
364:
360:
356:
351:
349:
345:
340:
338:
334:
330:
329:endomorphisms
326:
322:
318:
314:
310:
306:
302:
283:
280:
277:
274:
271:
264:
263:
262:
260:
244:
236:
233:of the group
232:
231:
220:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
141:
139:
135:
131:
127:
123:
119:
116:
112:
108:
104:
100:
96:
92:
88:
84:
70:
67:
59:
49:
45:
39:
38:
32:
27:
18:
17:
3901:
3896:Section 12.5
3889:
3884:Example 4.10
3877:
3865:
3833:
3827:
3800:
3794:
3772:
3750:
3732:
3710:
3694:, Springer,
3691:
3674:
3670:
3640:
3636:
3632:
3628:
3624:
3620:
3616:
3606:
3601:
3589:
3581:
3577:
3573:
3566:
3562:
3558:
3554:
3550:
3542:
3540:
3469:
3464:
3452:
3435:
3433:
3425:Consequences
3343:
3278:
2895:
2702:
2656:
2654:
2608:
2534:
2427:
2415:
2409:
2405:
2401:
2396:
2299:
2119:
2015:
2014:is called a
2011:
1989:
1943:
1812:
1808:
1804:
1802:
1618:
1518:in the form
1455:
1116:
1108:
1100:
1098:
958:
954:
946:
900:
776:
775:. The group
772:
769:Haar measure
722:
720:
709:
701:
697:
693:
689:
688:
680:
675:
671:
667:
663:
659:
655:
577:
573:
565:
561:
557:
553:
548:to be those
544:
540:
538:
536:
531:
527:
523:
519:
515:
511:
499:
495:
491:
487:
479:
475:
471:
469:
464:
460:
458:
454:circle group
447:
442:
436:
426:
422:
418:
416:
400:matrix group
395:
390:
384:
379:uniform norm
374:
370:
358:
354:
353:
341:
336:
332:
320:
316:
304:
300:
298:
258:
234:
228:
226:
216:
212:
208:
200:
196:
192:
188:
176:
172:
168:
160:
156:
152:
144:
142:
130:finite group
117:
107:Hermann Weyl
86:
80:
62:
53:
34:
3677:: 737–755,
3598:Élie Cartan
1990:A function
138:Issai Schur
111:Fritz Peter
83:mathematics
48:introducing
3918:Categories
3906:Knapp 2002
3872:Chapter 12
3692:Lie groups
3671:Math. Ann.
3663:References
3623:: Because
3035:of degree
2991:of degree
903:Knapp 1986
771:exists on
706:direct sum
683:Knapp 1986
412:Knapp 1986
404:Knapp 1986
348:Knapp 1986
309:continuous
56:March 2024
31:references
3894:Hall 2015
3882:Hall 2015
3870:Hall 2015
3739:EMS Press
3586:Lie group
3565:), where
3547:Hausdorff
3512:for some
3483:
3368:
3302:∈
3296:β
3290:α
3254:β
3238:α
3222:↦
3216:β
3210:α
3130:β
3110:α
2852:β
2829:α
2812:∈
2809:β
2803:α
2781:¯
2778:α
2771:β
2762:¯
2759:β
2754:−
2749:α
2721:
2589:θ
2567:θ
2515:θ
2493:θ
2473:π
2374:π
2348:π
2325:∑
2316:π
2312:χ
2270:π
2264:
2244:π
2240:χ
2216:π
2194:π
2190:χ
2180:character
2166:π
2043:−
1919:π
1908:≤
1896:≤
1885:Σ
1882:∈
1879:π
1875:∣
1867:π
1841:π
1803:for each
1785:⟩
1750:π
1747:⟨
1727:π
1681:π
1599:∗
1594:π
1586:⊗
1581:π
1570:Σ
1567:∈
1564:π
1558:^
1555:⨁
1503:×
1433:−
1419:π
1404:π
1395:⋅
1369:given by
1352:π
1341:
1315:×
1290:π
1260:π
1249:
1201:π
1190:π
1164:×
1080:π
1072:
1066:⊕
1061:π
1050:Σ
1047:∈
1044:π
1038:^
1035:⨁
872:−
834:ρ
819:given by
642:⟩
630:⟨
624:⟩
609:ρ
591:ρ
588:⟨
502:with the
408:Lie group
391:algebraic
344:Bump 2004
284:π
281:∘
272:φ
245:φ
97:that are
3858:Specific
3771:(2002),
3749:(1986),
3651:See also
2078:for all
700:. Then
654:for all
580:; i.e.,
572:for all
526:into GL(
486: :
346:, §4.1;
3850:2372901
3817:1970281
3741:, 2001
3594:theorem
3470:Theorem
2402:Theorem
2178:is the
1105:closure
568:) is a
562:unitary
449:actions
128:of any
103:abelian
99:compact
44:improve
3848:
3815:
3779:
3757:
3717:
3698:
3639:is an
3576:in GL(
2878:
2800:
2797:
821:acting
779:has a
520:∗(g,v)
319:, and
85:, the
33:, but
3846:JSTOR
3813:JSTOR
3609:limit
3436:basis
2705:SU(2)
2261:trace
783:ρ on
564:if ρ(
398:is a
373:) on
363:dense
323:is a
179:) of
3777:ISBN
3755:ISBN
3715:ISBN
3696:ISBN
3570:acts
3455:here
3169:and
3122:and
2098:and
670:→ U(
518:) =
459:Let
143:Let
136:and
3838:doi
3805:doi
3679:doi
3635:),
3596:of
3588:GL(
3557:in
3102:in
2707:as
2208:of
2118:in
2018:if
2010:on
1338:End
1246:End
1218:of
1069:dim
953:of
909:of
556:on
498:to
441:of
429:).
361:is
331:of
315:of
257:on
207:of
195:on
187:of
167:on
151:of
81:In
3920::
3844:,
3834:83
3832:,
3811:,
3801:73
3799:,
3737:,
3731:,
3675:97
3673:,
3647:.
3480:GL
3457:.
3450:.
3365:SU
3341:.
2718:SU
2420:.
729:,
658:,
514:)(
490:→
478:→
474:×
369:C(
311:)
227:A
140:.
3853:.
3840::
3807::
3786:.
3764:.
3724:.
3705:.
3686:.
3681::
3637:G
3633:G
3631:(
3629:L
3625:G
3621:G
3617:V
3602:G
3590:V
3582:G
3578:V
3574:G
3567:G
3563:G
3561:(
3559:L
3555:V
3551:G
3543:G
3532:.
3520:n
3500:)
3496:C
3492:;
3489:n
3486:(
3409:)
3404:3
3400:S
3396:(
3391:2
3387:L
3383:=
3380:)
3377:)
3374:2
3371:(
3362:(
3357:2
3353:L
3327:4
3322:R
3317:=
3312:2
3307:C
3299:)
3293:,
3287:(
3262:m
3258:)
3249:2
3245:z
3241:+
3233:1
3229:z
3225:(
3219:)
3213:,
3207:(
3182:2
3178:z
3155:1
3151:z
3090:m
3068:3
3064:S
3043:m
3019:m
2999:m
2975:m
2953:4
2948:R
2943:=
2938:2
2933:C
2909:3
2905:S
2881:,
2874:}
2870:1
2867:=
2862:2
2857:|
2848:|
2844:+
2839:2
2834:|
2825:|
2820:,
2816:C
2806:,
2794::
2789:)
2743:(
2737:{
2733:=
2730:)
2727:2
2724:(
2683:)
2680:G
2677:(
2672:2
2668:L
2657:G
2640:)
2635:1
2631:S
2627:(
2622:2
2618:L
2594:.
2586:n
2583:i
2579:e
2575:=
2572:)
2564:i
2560:e
2556:(
2551:n
2547:u
2520:.
2512:n
2509:i
2505:e
2501:=
2498:)
2490:i
2486:e
2482:(
2477:n
2447:1
2443:S
2439:=
2436:G
2412:.
2410:G
2406:G
2382:.
2377:)
2371:(
2366:i
2363:i
2359:u
2351:)
2345:(
2341:d
2335:1
2332:=
2329:i
2321:=
2285:.
2282:)
2279:)
2276:g
2273:(
2267:(
2258:=
2255:)
2252:g
2249:(
2146:)
2143:G
2140:(
2135:2
2131:L
2120:G
2106:h
2086:g
2066:)
2063:g
2060:(
2057:f
2054:=
2051:)
2046:1
2039:h
2035:g
2032:h
2029:(
2026:f
2012:G
1998:f
1971:.
1968:)
1965:G
1962:(
1957:2
1953:L
1928:}
1922:)
1916:(
1912:d
1905:j
1902:,
1899:i
1893:1
1888:,
1870:)
1864:(
1859:j
1856:i
1852:u
1844:)
1838:(
1834:d
1827:{
1813:d
1809:G
1805:g
1788:.
1780:i
1776:e
1772:,
1767:j
1763:e
1759:)
1756:g
1753:(
1744:=
1741:)
1738:g
1735:(
1730:)
1724:(
1719:j
1716:i
1712:u
1684:)
1678:(
1673:j
1670:i
1666:u
1643:)
1640:G
1637:(
1632:2
1628:L
1604:.
1590:E
1577:E
1548:=
1545:)
1542:G
1539:(
1534:2
1530:L
1506:G
1500:G
1480:)
1477:G
1474:(
1469:2
1465:L
1441:.
1436:1
1429:)
1425:h
1422:(
1416:A
1413:)
1410:g
1407:(
1401:=
1398:A
1392:)
1389:h
1386:,
1383:g
1380:(
1357:)
1348:E
1344:(
1318:G
1312:G
1286:E
1265:)
1256:E
1252:(
1226:G
1206:)
1197:E
1193:,
1187:(
1167:G
1161:G
1141:)
1138:G
1135:(
1130:2
1126:L
1112:π
1109:E
1101:G
1076:E
1057:E
1028:=
1025:)
1022:G
1019:(
1014:2
1010:L
985:)
982:G
979:(
974:2
970:L
959:G
957:(
955:L
947:G
933:)
930:G
927:(
922:2
918:L
886:.
883:)
880:g
875:1
868:h
864:(
861:f
858:=
855:)
852:g
849:(
846:f
843:)
840:h
837:(
807:)
804:G
801:(
796:2
792:L
777:G
773:G
754:)
751:G
748:(
743:2
739:L
723:G
712:.
710:G
702:H
698:H
694:G
676:C
672:H
668:G
664:H
660:w
656:v
639:w
636:,
633:v
627:=
621:w
618:)
615:g
612:(
606:,
603:v
600:)
597:g
594:(
578:G
574:g
566:g
558:H
554:G
545:H
541:G
532:H
528:H
524:G
516:v
512:g
510:(
508:∗
500:H
496:H
492:H
488:G
484:∗
480:H
476:H
472:G
465:H
461:G
443:G
427:G
425:(
423:L
419:G
402:(
396:G
381:.
375:G
371:G
359:G
337:V
333:V
321:L
317:G
305:V
301:G
278:L
275:=
259:G
235:G
217:G
213:G
211:(
209:L
201:G
199:(
197:L
193:G
189:G
177:G
175:(
173:L
169:G
161:G
159:(
157:C
153:G
145:G
120:(
118:G
69:)
63:(
58:)
54:(
40:.
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