38:
3026:
3006:
3014:
4529:
5057:
4322:
1748:
was one of the influential results of twentieth century mathematics. The combination of the PeterâWeyl theorem and the Weyl character formula led Weyl to a complete classification of the representations of a connected compact Lie group; this theory is described in the next section.
373:
Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers. (For information about the center, see the section below on fundamental group and center.)
4333:
1383:
In general, the center can be expressed in terms of the root lattice and the kernel of the exponential map for the maximal torus. The general method shows, for example, that the simply connected compact group corresponding to the exceptional root system
4883:
5434:
ranging over the dominant, analytically integral elementsâforms an orthonormal set in the space of square integrable class functions. But by the Weyl character formula, the characters of the irreducible representations form a subset of the
3960:
the representations have been classified. In Weyl's analysis of the compact group case, however, the Weyl character formula is actually a crucial part of the classification itself. Specifically, in Weyl's analysis of the representations of
2449:
3965:, the hardest part of the theoremâshowing that every dominant, analytically integral element is actually the highest weight of some representationâis proved in a totally different way from the usual Lie algebra construction using
4077:
4179:
3794:
2986:
1956:
1768:
are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(Ï) must itself be a Lie subgroup in the unitary group. If
2615:
3862:
2779:
357:
2131:
822:. These structures then play an essential role both in the classification of connected compact groups (described above) and in the representation theory of a fixed such group (described below).
543:
4742:. We focus on the most difficult part of the theorem, showing that every dominant, analytically integral element is the highest weight of some (finite-dimensional) irreducible representation.
3656:
493:
443:
3720:
2218:
4618:
each with multiplicity one. (The weights are the integers appearing in the exponents of the exponentials and the multiplicities are the coefficients of the exponentials.) Since there are
2883:
935:
4524:{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta }+e^{i(m-2)\theta }+\cdots e^{-i(m-2)\theta }+e^{-im\theta }.}
2018:
1060:
4613:
3294:
711:
5462:'s. And by the PeterâWeyl theorem, the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions. If there were some
3425:
5169:
norm of the character is 1. Specifically, if there were any additional terms in the numerator, the Weyl integral formula would force the norm of the character to be greater than 1.
1354:
1259:
1227:
1163:
996:
859:
1195:
2472:
3451:
5682:
5569:
5538:
5507:
5460:
5412:
5382:
5335:
5248:
5197:
3251:
2667:
3590:
3342:
2265:
2045:
2301:
1817:
5052:{\displaystyle \mathrm {X} (e^{H})={\frac {\sum _{w\in W}\det(w)e^{i\langle w\cdot (\lambda +\rho ),H\rangle }}{\sum _{w\in W}\det(w)e^{i\langle w\cdot \rho ,H\rangle }}}}
4855:
3928:
3077:
2732:
2640:
1698:
808:
5589:
5480:
5432:
5355:
5308:
5219:
4833:
3538:
3386:
3217:
3173:
3153:
2842:
2660:
2559:
2519:
2321:
2241:
5677:
3558:
3492:
3475:
3362:
3241:
3193:
3121:
3097:
3051:
3000:
2372:
1821:
892:
2380:
2075:
5167:
5120:
2702:
2539:
2499:
2352:
2154:
1983:
1879:
1497:
1468:
1438:
1409:
1087:
1023:
962:
673:
4875:
4813:
3684:
4668:
4642:
2802:
5288:
5268:
5140:
5100:
5080:
4785:
4740:
4720:
4700:
4167:
4144:
4124:
4104:
3902:
3882:
2903:
2822:
1718:
1649:
1629:
1378:
1322:
1302:
1279:
775:
751:
731:
614:
is semisimple. Conversely, every complex semisimple Lie algebra has a compact real form isomorphic to the Lie algebra of a compact, simply connected Lie group.
5357:
is the highest weight of a representation; nevertheless, the expressions on the right-hand side of the character formula gives a well-defined set of functions
3565:
1324:. (The group SO(2) is an exceptionâthe center is the whole group, even though most elements are not roots of the identity.) Thus, for example, the center of
4003:
4317:{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {\sin((m+1)\theta )}{\sin(\theta )}}.}
1604:, and Haar measure of a coset will be the reciprocal of the index. Therefore, integrals are often computable quite directly, a fact applied constantly in
3600:
is simply connected, the set of possible highest weights in the group sense is the same as the set of possible highest weights in the Lie algebra sense.
237:
this exhausts the list of examples (which already includes some redundancies). This classification is described in more detail in the next subsection.
115:
form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include
4767:, which states that the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions on
1809:
3728:
369:: Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus.
2915:
4086:, the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers
3937:
The study of characters is an important part of the representation theory of compact groups. One crucial result, which is a corollary of the
1891:
825:
The root systems associated to the simple compact groups appearing in the classification of simply connected compact groups are as follows:
5122:
is half the sum of the positive roots. (The notation uses the convention of "real weights"; this convention requires an explicit factor of
4534:(If we use the formula for the sum of a finite geometric series on the above expression and simplify, we obtain the earlier expression.)
1808:
Certain simple examples of the representation theory of compact Lie groups can be worked out by hand, such as the representations of the
3956:
In the closely related representation theory of semisimple Lie algebras, the Weyl character formula is an additional result established
2567:
17:
3297:
1797:
781:
81:(when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of
4670:. Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation.
5544:(the set of characters of the irreducible representations) would be contained in a strictly larger orthonormal set (the set of
4538:
3561:
3517:
4764:
5782:
Peter, F.; Weyl, H. (1927), "Die VollstÀndigkeit der primitiven
Darstellungen einer geschlossenen kontinuierlichen Gruppe",
3803:
2737:
1728:
The representation theory of compact groups (not necessarily Lie groups and not necessarily connected) was founded by the
295:
3997:, but we here look at them from the group point of view. We take the maximal torus to be the set of matrices of the form
2083:
89:
and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to
503:
5939:
3617:
453:
403:
5957:
1700:
as an orthogonal direct sum of finite-dimensional subspaces of matrix entries for the irreducible representations of
3689:
3560:
are analytically integral in the sense described in the previous subsection. Every analytically integral element is
3250:
We now briefly describe the structures needed to formulate the theorem; more details can be found in the article on
2162:
1796:
The unknown part of the representation theory of compact groups is thereby, roughly speaking, thrown back onto the
1442:
is one of very few simple compact groups that are simultaneously simply connected and center free. (The others are
4170:
5632:
2847:
899:
1125:
It is important to know whether a connected compact Lie group is simply connected, and if not, to determine its
3994:
3487:
3123:
is one-dimensional. Nevertheless, the restriction decomposes as a direct sum of irreducible representations of
1988:
1813:
1030:
5142:
in the exponent.) Weyl's proof of the character formula is analytic in nature and hinges on the fact that the
4547:
3269:
678:
3395:
1304:
can easily be computed "by hand," and in most cases consists simply of whatever roots of the identity are in
3508:
every dominant, analytically integral element arises as the highest weight of an irreducible representation.
1327:
1232:
1200:
1136:
969:
832:
1777:, for the kernel of Ï smaller and smaller, of finite-dimensional unitary representations, which identifies
1168:
5337:
form an orthonormal family of class functions. We emphasize that we do not currently know that every such
4791:
With these tools in hand, we proceed with the proof. The first major step in the argument is to prove the
1284:
It is also important to know the center of a connected compact Lie group. The center of a classical group
5975:
2457:
591:. For each of these groups, the center is known explicitly. The classification is through the associated
3430:
1592:
Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to
602:
The classification of compact, simply connected Lie groups is the same as the classification of complex
3993:
We now consider the case of the compact group SU(2). The representations are often considered from the
3941:, is that the characters form an orthonormal basis for the set of square-integrable class functions in
90:
5547:
5516:
5485:
5438:
5390:
5360:
5313:
5226:
5175:
3516:
is then almost the same as for semisimple Lie algebras, with one notable exception: The concept of an
5985:
5600:
3571:
3323:
2246:
2026:
1543:
provides a large supply of examples of compact commutative groups. These are in duality with abelian
3949:, which gives an explicit formula for the characterâor, rather, the restriction of the character to
3564:
in the Lie algebra sense, but not the other way around. (This phenomenon reflects that, in general,
2273:
5624:
4757:
145:
4838:
4678:
We now outline the proof of the theorem of the highest weight, following the original argument of
3911:
3060:
2707:
2623:
3018:
1786:
1753:
1658:
3001:
Lie algebra representation § Classifying finite-dimensional representations of Lie algebras
1130:
1120:
787:
5644:
4792:
3974:
3970:
3946:
3938:
3609:
2444:{\displaystyle \Gamma =\left\{H\in {\mathfrak {t}}\mid e^{2\pi H}=\operatorname {Id} \right\},}
1761:
1745:
1729:
1652:
1556:
1133:
to computing the fundamental group. The first approach applies to the classical compact groups
819:
603:
226:
5615:
The influence of the compact group theory on non-compact groups was formulated by Weyl in his
5574:
5465:
5417:
5340:
5293:
5204:
4818:
3523:
3371:
3202:
3158:
3138:
2827:
2645:
2544:
2504:
2306:
2226:
5659:
5599:
The topic of recovering a compact group from its representation theory is the subject of the
3543:
3460:
3347:
3226:
3178:
3106:
3082:
3036:
2357:
864:
448:
398:
183:
172:
94:
2054:
1281:. The second approach uses the root system and applies to all connected compact Lie groups.
5620:
5145:
5105:
2680:
2524:
2481:
2330:
2139:
1968:
1864:
1475:
1446:
1416:
1387:
1065:
1001:
940:
675:
and that is not contained in any larger subgroup of this type. A basic example is the case
651:
4860:
4798:
3669:
8:
5980:
4647:
4621:
3457:
than another if their difference can be expressed as a linear combination of elements of
1586:
1534:
610:
is a simply connected compact Lie group, then the complexification of the Lie algebra of
4750:
2784:
5604:
5273:
5253:
5125:
5085:
5065:
4770:
4725:
4705:
4685:
4152:
4129:
4109:
4089:
3887:
3867:
2888:
2807:
1703:
1634:
1614:
1540:
1363:
1307:
1287:
1264:
760:
736:
716:
250:
4072:{\displaystyle {\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}.}
5953:
5935:
1597:
1126:
86:
70:
1091:
The exceptional compact Lie groups correspond to the five exceptional root systems G
5791:
1858:
1737:
1582:
1472:
1443:
1412:
577:
570:
563:
556:
549:
382:
215:
208:
201:
194:
187:
137:
5199:
denote the function on the right-hand side of the character formula. We show that
5664:
5616:
2781:
and the kernel of the (scaled) exponential map is the set of numbers of the form
1526:
587:
are to avoid special isomorphisms among the various families for small values of
260:
230:
101:
5509:
would not be the character of a representation. Thus, the characters would be a
3505:
two irreducible representations with the same highest weight are isomorphic, and
2666:. This integrality condition is related to, but not identical to, the notion of
1773:
is not itself a Lie group, there must be a kernel to Ï. Further one can form an
5628:
5310:
ranges over the set of dominant, analytically integral elements, the functions
1774:
1574:
1544:
1522:
1508:
Amongst groups that are not Lie groups, and so do not carry the structure of a
596:
264:
234:
4082:
According to the example discussed above in the section on representations of
3300:. Specifically, the weights are the nonzero weights for the adjoint action of
1800:. This theory is rather rich in detail, but is qualitatively well understood.
5969:
5649:
3025:
2734:
of absolute value 1. The Lie algebra is the set of purely imaginary numbers,
1832:
1782:
1741:
1605:
1593:
623:
164:
78:
46:
5762:
4126:, there is a unique irreducible representation of SU(2) with highest weight
3789:{\displaystyle \mathrm {X} (x)=\operatorname {trace} (\Pi (x)),\quad x\in K}
5482:
that is not the highest weight of a representation, then the corresponding
4679:
3966:
1733:
1601:
1570:
1566:
1562:
1530:
780:
The maximal torus in a compact group plays a role analogous to that of the
362:
Meanwhile, for connected compact Lie groups, we have the following result:
120:
82:
31:
5769:, Actualités Scientifiques et Industrielles, vol. 869, Paris: Hermann
4644:
weights, each with multiplicity 1, the dimension of the representation is
3247:
is to classify the irreducible representations in terms of their weights.
2981:{\displaystyle \rho (e^{i\theta })=e^{ik\theta },\quad k\in \mathbb {Z} .}
2844:
takes integer values on all such numbers if and only if it is of the form
5934:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
5654:
3313:
1803:
811:
784:
in a complex semisimple Lie algebra. In particular, once a maximal torus
627:
592:
127:
54:
5932:
Lie Groups, Lie
Algebras, and Representations An Elementary Introduction
5795:
3502:
the highest weight is always a dominant, analytically integral element,
1951:{\displaystyle \rho :T\rightarrow GL(1;\mathbb {C} )=\mathbb {C} ^{*}.}
1820:. We focus here on the general theory. See also the parallel theory of
815:
498:
153:
3029:
Black dots indicate the dominant integral elements for the group SU(3)
37:
5627:, and the representation theory of such groups, developed largely by
112:
1756:
gives a survey of the whole representation theory of compact groups
777:
belongs to a maximal torus and that all maximal tori are conjugate.
5672:
5635:
to such a subgroup, and also the model of Weyl's character theory.
4327:
We can also write the character as sum of exponentials as follows:
4169:
is encoded in its character. Now, the Weyl character formula says,
4149:
Much information about the representation corresponding to a given
3131:
may occur more than once.) Now, each irreducible representation of
1509:
595:(for a fixed maximal torus), which in turn are classified by their
381:
is a product of finitely many compact, connected, simply-connected
74:
1565:, which will be invariant by both left and right translation (the
30:
This article is about mathematics. For astronomy of galaxies, see
1596:, and in the Lie group cases can always be given by an invariant
3491:, which is closely related to the analogous theorem classifying
3175:
occurs at least once in the decomposition of the restriction of
5767:
L'intégration dans les groupes topologiques et ses applications
5683:
Weights in the representation theory of semisimple Lie algebras
2610:{\displaystyle \lambda (H)\in \mathbb {Z} ,\quad H\in \Gamma ,}
42:
5678:
Classifying finite-dimensional representations of Lie algebras
4541:, we can read off that the weights of the representation are
3013:
1827:
Throughout this section, we fix a connected compact Lie group
1360:
th roots of unity times the identity, a cyclic group of order
377:
Finally, every compact, connected, simply-connected Lie group
5540:'s. But then we have an impossible situation: an orthonormal
5290:. Then using the Weyl integral formula, one can show that as
3009:
Example of the weights of a representation of the group SU(3)
393:
each of which is isomorphic to exactly one of the following:
3005:
4537:
From this last expression and the standard formula for the
3800:
This function is easily seen to be a class function, i.e.,
1793:
is found is another consequence of the PeterâWeyl theorem.
3053:
denote a finite-dimensional irreducible representation of
5591:
must actually be the highest weight of a representation.
5221:
is not known to be the highest weight of a representation
3981:
are realized inside the space of continuous functions on
3512:
The theorem of the highest weight for representations of
1785:
of compact Lie groups. Here the fact that in the limit a
632:
A key idea in the study of a connected compact Lie group
5925:, Graduate Texts in Mathematics, vol. 98, Springer
4539:
character in terms of the weights of the representation
3969:. In Weyl's approach, the construction is based on the
3953:âin terms of the highest weight of the representation.
4351:
4197:
4012:
3857:{\displaystyle \mathrm {X} (xyx^{-1})=\mathrm {X} (y)}
3480:
The irreducible finite-dimensional representations of
3021:" representation of SU(3), as used in particle physics
1804:
Representation theory of a connected compact Lie group
5610:
5577:
5550:
5519:
5488:
5468:
5441:
5420:
5393:
5363:
5343:
5316:
5296:
5276:
5256:
5229:
5207:
5178:
5148:
5128:
5108:
5088:
5068:
4886:
4863:
4841:
4821:
4815:
is an irreducible representation with highest weight
4801:
4773:
4728:
4708:
4688:
4650:
4624:
4550:
4336:
4182:
4155:
4132:
4112:
4092:
4006:
3914:
3890:
3870:
3806:
3731:
3692:
3672:
3620:
3574:
3546:
3526:
3463:
3433:
3398:
3374:
3350:
3326:
3272:
3229:
3205:
3181:
3161:
3141:
3109:
3085:
3063:
3039:
2918:
2891:
2850:
2830:
2810:
2787:
2774:{\displaystyle H=i\theta ,\,\theta \in \mathbb {R} ,}
2740:
2710:
2683:
2648:
2626:
2570:
2547:
2527:
2507:
2501:
in order to avoid such factors elsewhere.) Then for
2484:
2460:
2383:
2360:
2333:
2309:
2276:
2249:
2229:
2165:
2142:
2086:
2057:
2029:
2023:
To describe these representations concretely, we let
1991:
1971:
1894:
1867:
1760:. That is, by the PeterâWeyl theorem the irreducible
1706:
1661:
1637:
1617:
1478:
1449:
1419:
1390:
1366:
1330:
1310:
1290:
1267:
1235:
1203:
1171:
1139:
1068:
1033:
1004:
972:
943:
902:
867:
835:
790:
763:
739:
719:
681:
654:
648:
that is isomorphic to a product of several copies of
506:
456:
406:
298:
3499:
every irreducible representation has highest weight,
2478:. (We scale the exponential map here by a factor of
1600:. In the profinite case there are many subgroups of
352:{\displaystyle 1\to G_{0}\to G\to \pi _{0}(G)\to 1.}
3127:. (Note that a given irreducible representation of
2303:is not injective, not every such linear functional
2126:{\displaystyle h=e^{H},\quad H\in {\mathfrak {t}}.}
1533:are compact groups, a basic fact for the theory of
49:
is a compact Lie group with complex multiplication.
5583:
5563:
5532:
5501:
5474:
5454:
5426:
5406:
5376:
5349:
5329:
5302:
5282:
5262:
5242:
5213:
5191:
5161:
5134:
5114:
5094:
5074:
5051:
4869:
4849:
4827:
4807:
4779:
4734:
4714:
4694:
4662:
4636:
4607:
4523:
4316:
4161:
4138:
4118:
4098:
4071:
3922:
3896:
3876:
3856:
3788:
3714:
3678:
3650:
3584:
3552:
3532:
3469:
3445:
3419:
3380:
3356:
3336:
3288:
3235:
3211:
3187:
3167:
3147:
3115:
3091:
3071:
3045:
2980:
2897:
2877:
2836:
2816:
2796:
2773:
2726:
2696:
2654:
2634:
2609:
2553:
2533:
2513:
2493:
2466:
2443:
2366:
2346:
2315:
2295:
2259:
2235:
2212:
2148:
2125:
2069:
2039:
2012:
1977:
1950:
1873:
1712:
1692:
1643:
1623:
1581:, Ă), and so 1). In other words, these groups are
1491:
1462:
1432:
1403:
1372:
1348:
1316:
1296:
1273:
1253:
1221:
1189:
1157:
1081:
1054:
1017:
990:
956:
929:
886:
853:
802:
769:
745:
725:
705:
667:
538:{\displaystyle \operatorname {Spin} (n),\,n\geq 7}
537:
487:
437:
351:
5270:, which therefore extends to a class function on
4106:. The general theory then tells us that for each
3977:. Ultimately, the irreducible representations of
2909:in this case are one-dimensional and of the form
289:is compact. We therefore have a finite extension
5967:
5920:
5713:
5003:
4930:
3651:{\displaystyle \Pi :K\to \operatorname {GL} (V)}
3298:construction for complex semisimple Lie algebras
617:
488:{\displaystyle \operatorname {SU} (n),\,n\geq 3}
438:{\displaystyle \operatorname {Sp} (n),\,n\geq 1}
3243:. The strategy of the representation theory of
1114:
229:of compact Lie groups states that up to finite
100:In the following we will assume all groups are
5250:is a well-defined, Weyl-invariant function on
3715:{\displaystyle \mathrm {X} :K\to \mathbb {C} }
1861:tells us that each irreducible representation
1740:of the compact connected Lie groups, based on
5947:
3603:
3103:. This restriction is not irreducible unless
2991:
2213:{\displaystyle \rho (e^{H})=e^{i\lambda (H)}}
1845:
5948:Hofmann, Karl H.; Morris, Sidney A. (1998),
5041:
5023:
4980:
4950:
2642:is the set of integers. A linear functional
1585:. Haar measure is easily normalized to be a
5921:Bröcker, Theodor; tom Dieck, Tammo (1985),
4745:The tools for the proof are the following:
3493:representations of a semisimple Lie algebra
3155:as in the preceding subsection. If a given
2878:{\displaystyle \lambda (i\theta )=k\theta }
2670:in the setting of semisimple Lie algebras.
1822:representations of a semisimple Lie algebra
930:{\displaystyle \operatorname {Spin} (2n+1)}
4673:
2374:denote the kernel of the exponential map:
5781:
5387:Now comes the conclusion. The set of all
3708:
3592:comes from a representation of the group
3065:
2971:
2764:
2756:
2628:
2587:
2013:{\displaystyle S^{1}\subset \mathbb {C} }
2006:
1935:
1923:
1525:, and constructions from it. In fact any
1055:{\displaystyle \operatorname {Spin} (2n)}
525:
475:
425:
4608:{\displaystyle m,m-2,\ldots ,-(m-2),-m,}
3289:{\displaystyle R\subset {\mathfrak {t}}}
3266:). The construction of this root system
3024:
3012:
3004:
1798:complex representations of finite groups
1723:
733:to be the group of diagonal elements in
706:{\displaystyle K=\operatorname {SU} (n)}
36:
5384:, and these functions are orthonormal.
3420:{\displaystyle \lambda (\alpha )\geq 0}
3079:). We then consider the restriction of
2662:satisfying this condition is called an
27:Topological group with compact topology
14:
5968:
4765:PeterâWeyl theorem for class functions
1349:{\displaystyle \operatorname {SU} (n)}
1254:{\displaystyle \operatorname {Sp} (n)}
1222:{\displaystyle \operatorname {SO} (n)}
1158:{\displaystyle \operatorname {SU} (n)}
991:{\displaystyle \operatorname {Sp} (n)}
854:{\displaystyle \operatorname {SU} (n)}
548:or one of the five exceptional groups
5923:Representations of Compact Lie Groups
4702:be a connected compact Lie group and
3453:. Finally, we say that one weight is
2905:. The irreducible representations of
1190:{\displaystyle \operatorname {U} (n)}
107:
5929:
5904:
5892:
5880:
5868:
5856:
5844:
5832:
5820:
5808:
5761:
5749:
5737:
5725:
5701:
3930:is determined by its restriction to
3368:and we say that an integral element
3135:is described by a linear functional
2323:gives rise to a well-defined map of
1651:is the associated Haar measure, the
1589:, analogous to dΞ/2Ï on the circle.
1529:is a compact group. This means that
1129:. For compact Lie groups, there are
3577:
3329:
3304:on the complexified Lie algebra of
3281:
3262:(relative to a given maximal torus
2824:is an integer. A linear functional
2467:{\displaystyle \operatorname {Id} }
2403:
2252:
2115:
2032:
1503:
1121:Fundamental group § Lie groups
757:which states that every element of
24:
5611:From compact to non-compact groups
5552:
5521:
5490:
5443:
5395:
5365:
5318:
5231:
5180:
4888:
4864:
4843:
4802:
4338:
4184:
3916:
3841:
3808:
3758:
3733:
3694:
3673:
3621:
3547:
3464:
3446:{\displaystyle \alpha \in \Delta }
3440:
3351:
3312:has all the usual properties of a
3230:
3182:
3110:
3086:
3040:
2601:
2384:
2361:
1512:, examples are the additive group
1172:
810:has been chosen, one can define a
25:
5997:
4173:, that the character is given by
3988:
1752:A combination of Weyl's work and
240:
5564:{\displaystyle \Phi _{\lambda }}
5533:{\displaystyle \Phi _{\lambda }}
5502:{\displaystyle \Phi _{\lambda }}
5455:{\displaystyle \Phi _{\lambda }}
5407:{\displaystyle \Phi _{\lambda }}
5377:{\displaystyle \Phi _{\lambda }}
5330:{\displaystyle \Phi _{\lambda }}
5243:{\displaystyle \Phi _{\lambda }}
5192:{\displaystyle \Phi _{\lambda }}
3477:with non-negative coefficients.
3252:weights in representation theory
1537:in the case of infinite degree.
45:of center 0 and radius 1 in the
5950:The structure of compact groups
5914:
5898:
5886:
5874:
5862:
5850:
5838:
5826:
5633:restriction of a representation
5603:, now often recast in terms of
3776:
3585:{\displaystyle {\mathfrak {k}}}
3337:{\displaystyle {\mathfrak {t}}}
2963:
2594:
2270:Now, since the exponential map
2260:{\displaystyle {\mathfrak {t}}}
2106:
2040:{\displaystyle {\mathfrak {t}}}
1550:
5814:
5802:
5775:
5755:
5743:
5731:
5719:
5707:
5695:
5012:
5006:
4971:
4959:
4939:
4933:
4905:
4892:
4590:
4578:
4488:
4476:
4451:
4439:
4305:
4299:
4288:
4282:
4270:
4267:
3851:
3845:
3834:
3812:
3770:
3767:
3761:
3755:
3743:
3737:
3704:
3645:
3639:
3630:
3408:
3402:
3316:, except that the elements of
2938:
2922:
2863:
2854:
2580:
2574:
2296:{\displaystyle H\mapsto e^{H}}
2280:
2205:
2199:
2182:
2169:
1927:
1913:
1904:
1687:
1672:
1343:
1337:
1248:
1242:
1216:
1210:
1184:
1178:
1152:
1146:
1062:correspond to the root system
1049:
1040:
998:correspond to the root system
985:
979:
966:The compact symplectic groups
937:correspond to the root system
924:
909:
861:correspond to the root system
848:
842:
700:
694:
519:
513:
469:
463:
419:
413:
343:
340:
334:
321:
315:
302:
13:
1:
5716:, Chapter V, Sections 7 and 8
5688:
4795:. The formula states that if
3973:and an analytic proof of the
3945:. A second key result is the
2664:analytically integral element
1736:went on to give the detailed
1261:and proceeds by induction on
618:Maximal tori and root systems
285:) which must be finite since
5714:Bröcker & tom Dieck 1985
4850:{\displaystyle \mathrm {X} }
3923:{\displaystyle \mathrm {X} }
3072:{\displaystyle \mathbb {C} }
2727:{\displaystyle e^{i\theta }}
2635:{\displaystyle \mathbb {Z} }
1655:provides a decomposition of
1115:Fundamental group and center
818:similar to what one has for
713:, in which case we may take
277:is the group of components Ï
245:Given any compact Lie group
7:
5638:
2521:to give a well-defined map
2474:is the identity element of
2223:for some linear functional
1818:special unitary group SU(3)
1814:special unitary group SU(2)
1693:{\displaystyle L^{2}(K,dm)}
1561:Compact groups all carry a
829:The special unitary groups
10:
6002:
5594:
3607:
3604:The Weyl character formula
3520:is different. The weights
3254:. We need the notion of a
2998:
1554:
1411:has trivial center. Thus,
1118:
803:{\displaystyle T\subset K}
621:
29:
4722:a fixed maximal torus in
3995:Lie algebra point of view
3596:.) On the other hand, if
3495:. The result says that:
3484:are then classified by a
2992:Representation theory of
1846:Representation theory of
182:the compact forms of the
79:compact topological space
18:Compact topological group
5625:maximal compact subgroup
5584:{\displaystyle \lambda }
5475:{\displaystyle \lambda }
5427:{\displaystyle \lambda }
5350:{\displaystyle \lambda }
5303:{\displaystyle \lambda }
5214:{\displaystyle \lambda }
4828:{\displaystyle \lambda }
3566:not every representation
3533:{\displaystyle \lambda }
3381:{\displaystyle \lambda }
3344:. We then choose a base
3212:{\displaystyle \lambda }
3168:{\displaystyle \lambda }
3148:{\displaystyle \lambda }
2837:{\displaystyle \lambda }
2655:{\displaystyle \lambda }
2554:{\displaystyle \lambda }
2514:{\displaystyle \lambda }
2316:{\displaystyle \lambda }
2236:{\displaystyle \lambda }
753:. A basic result is the
399:compact symplectic group
146:special orthogonal group
5930:Hall, Brian C. (2015),
5631:, uses intensively the
4674:An outline of the proof
3553:{\displaystyle \Sigma }
3486:theorem of the highest
3470:{\displaystyle \Delta }
3357:{\displaystyle \Delta }
3296:is very similar to the
3236:{\displaystyle \Sigma }
3188:{\displaystyle \Sigma }
3116:{\displaystyle \Sigma }
3092:{\displaystyle \Sigma }
3046:{\displaystyle \Sigma }
2367:{\displaystyle \Gamma }
1985:must actually map into
1787:faithful representation
1762:unitary representations
1631:is a compact group and
887:{\displaystyle A_{n-1}}
820:semisimple Lie algebras
604:semisimple Lie algebras
5952:, Berlin: de Gruyter,
5907:Sections 12.4 and 12.5
5895:Sections 12.4 and 12.5
5585:
5565:
5534:
5503:
5476:
5456:
5428:
5408:
5378:
5351:
5331:
5304:
5284:
5264:
5244:
5215:
5193:
5163:
5136:
5116:
5096:
5082:in the Lie algebra of
5076:
5053:
4871:
4851:
4829:
4809:
4793:Weyl character formula
4781:
4736:
4716:
4696:
4664:
4638:
4609:
4525:
4318:
4163:
4140:
4120:
4100:
4073:
3975:Weyl character formula
3947:Weyl character formula
3924:
3898:
3878:
3858:
3790:
3716:
3680:
3652:
3610:Weyl character formula
3586:
3554:
3534:
3471:
3447:
3421:
3382:
3358:
3338:
3290:
3237:
3213:
3189:
3169:
3149:
3117:
3093:
3073:
3047:
3030:
3022:
3010:
2982:
2899:
2879:
2838:
2818:
2798:
2775:
2728:
2698:
2673:Suppose, for example,
2656:
2636:
2611:
2555:
2535:
2515:
2495:
2468:
2445:
2368:
2348:
2317:
2297:
2261:
2237:
2214:
2150:
2127:
2071:
2070:{\displaystyle h\in T}
2047:be the Lie algebra of
2041:
2014:
1979:
1952:
1875:
1746:Weyl character formula
1744:theory. The resulting
1714:
1694:
1645:
1625:
1493:
1464:
1434:
1405:
1374:
1350:
1318:
1298:
1275:
1255:
1223:
1191:
1159:
1083:
1056:
1019:
992:
958:
931:
888:
855:
804:
771:
747:
727:
707:
669:
583:. The restrictions on
539:
489:
439:
353:
227:classification theorem
184:exceptional Lie groups
50:
5660:Locally compact group
5601:TannakaâKrein duality
5586:
5566:
5535:
5513:subset of the set of
5504:
5477:
5457:
5429:
5409:
5379:
5352:
5332:
5305:
5285:
5265:
5245:
5216:
5194:
5164:
5162:{\displaystyle L^{2}}
5137:
5117:
5115:{\displaystyle \rho }
5097:
5077:
5054:
4872:
4852:
4835:, then the character
4830:
4810:
4782:
4758:Weyl integral formula
4737:
4717:
4697:
4682:. We continue to let
4665:
4639:
4610:
4526:
4319:
4164:
4141:
4121:
4101:
4074:
3925:
3899:
3879:
3859:
3791:
3717:
3681:
3658:is representation of
3653:
3587:
3555:
3535:
3472:
3448:
3422:
3383:
3359:
3339:
3291:
3238:
3214:
3190:
3170:
3150:
3118:
3094:
3074:
3048:
3028:
3016:
3008:
2983:
2900:
2880:
2839:
2819:
2799:
2776:
2729:
2699:
2697:{\displaystyle S^{1}}
2657:
2637:
2612:
2556:
2536:
2534:{\displaystyle \rho }
2516:
2496:
2494:{\displaystyle 2\pi }
2469:
2446:
2369:
2349:
2347:{\displaystyle S^{1}}
2318:
2298:
2262:
2238:
2215:
2151:
2149:{\displaystyle \rho }
2136:In such coordinates,
2128:
2072:
2042:
2015:
1980:
1978:{\displaystyle \rho }
1953:
1876:
1874:{\displaystyle \rho }
1724:Representation theory
1715:
1695:
1646:
1626:
1569:must be a continuous
1494:
1492:{\displaystyle E_{8}}
1465:
1463:{\displaystyle F_{4}}
1435:
1433:{\displaystyle G_{2}}
1406:
1404:{\displaystyle G_{2}}
1375:
1351:
1319:
1299:
1276:
1256:
1224:
1192:
1160:
1084:
1082:{\displaystyle D_{n}}
1057:
1027:The even spin groups
1020:
1018:{\displaystyle C_{n}}
993:
959:
957:{\displaystyle B_{n}}
932:
889:
856:
805:
772:
748:
728:
708:
670:
668:{\displaystyle S^{1}}
640:, that is a subgroup
540:
490:
449:special unitary group
440:
354:
173:special unitary group
95:representation theory
40:
5621:semisimple Lie group
5575:
5548:
5517:
5486:
5466:
5439:
5418:
5391:
5361:
5341:
5314:
5294:
5274:
5254:
5227:
5205:
5176:
5146:
5126:
5106:
5086:
5066:
4884:
4870:{\displaystyle \Pi }
4861:
4839:
4819:
4808:{\displaystyle \Pi }
4799:
4771:
4726:
4706:
4686:
4648:
4622:
4548:
4334:
4180:
4153:
4130:
4110:
4090:
4004:
3912:
3888:
3868:
3804:
3729:
3690:
3679:{\displaystyle \Pi }
3670:
3618:
3572:
3544:
3540:of a representation
3524:
3461:
3431:
3396:
3372:
3348:
3324:
3270:
3227:
3203:
3179:
3159:
3139:
3107:
3083:
3061:
3037:
2916:
2889:
2848:
2828:
2808:
2785:
2738:
2708:
2681:
2646:
2624:
2568:
2545:
2525:
2505:
2482:
2458:
2381:
2358:
2331:
2307:
2274:
2247:
2227:
2163:
2140:
2084:
2055:
2051:and we write points
2027:
1989:
1969:
1892:
1885:is one-dimensional:
1865:
1810:rotation group SO(3)
1704:
1659:
1635:
1615:
1535:algebraic extensions
1476:
1447:
1417:
1388:
1364:
1328:
1308:
1288:
1265:
1233:
1201:
1169:
1137:
1131:two basic approaches
1066:
1031:
1002:
970:
941:
900:
896:The odd spin groups
865:
833:
788:
761:
737:
717:
679:
652:
636:is the concept of a
504:
454:
404:
296:
5619:. Inside a general
4663:{\displaystyle m+1}
4637:{\displaystyle m+1}
3686:to be the function
3568:of the Lie algebra
2704:of complex numbers
2156:will have the form
1587:probability measure
152:) and its covering
5976:Topological groups
5796:10.1007/BF01447892
5645:PeterâWeyl theorem
5605:Tannakian category
5581:
5561:
5530:
5499:
5472:
5452:
5424:
5404:
5374:
5347:
5327:
5300:
5280:
5260:
5240:
5211:
5189:
5159:
5132:
5112:
5092:
5072:
5049:
5002:
4929:
4867:
4847:
4825:
4805:
4777:
4732:
4712:
4692:
4660:
4634:
4605:
4521:
4399:
4314:
4245:
4159:
4136:
4116:
4096:
4069:
4060:
3971:PeterâWeyl theorem
3939:PeterâWeyl theorem
3920:
3894:
3874:
3854:
3786:
3712:
3676:
3648:
3582:
3550:
3530:
3467:
3443:
3417:
3378:
3354:
3334:
3308:. The root system
3286:
3233:
3209:
3185:
3165:
3145:
3113:
3089:
3069:
3043:
3031:
3023:
3011:
2978:
2895:
2875:
2834:
2814:
2797:{\displaystyle in}
2794:
2771:
2724:
2694:
2677:is just the group
2652:
2632:
2607:
2551:
2531:
2511:
2491:
2464:
2441:
2364:
2344:
2313:
2293:
2257:
2233:
2210:
2146:
2123:
2067:
2037:
2010:
1975:
1948:
1871:
1730:PeterâWeyl theorem
1710:
1690:
1653:PeterâWeyl theorem
1641:
1621:
1557:PeterâWeyl theorem
1541:Pontryagin duality
1489:
1460:
1430:
1401:
1370:
1346:
1314:
1294:
1271:
1251:
1219:
1187:
1155:
1079:
1052:
1015:
988:
954:
927:
884:
851:
800:
767:
743:
723:
703:
665:
535:
485:
435:
349:
251:identity component
108:Compact Lie groups
51:
5571:'s). Thus, every
5283:{\displaystyle K}
5263:{\displaystyle T}
5135:{\displaystyle i}
5095:{\displaystyle T}
5075:{\displaystyle H}
5047:
4987:
4914:
4780:{\displaystyle K}
4735:{\displaystyle K}
4715:{\displaystyle T}
4695:{\displaystyle K}
4309:
4162:{\displaystyle m}
4139:{\displaystyle m}
4119:{\displaystyle m}
4099:{\displaystyle m}
3897:{\displaystyle y}
3877:{\displaystyle x}
2898:{\displaystyle k}
2885:for some integer
2817:{\displaystyle n}
1713:{\displaystyle K}
1644:{\displaystyle m}
1624:{\displaystyle K}
1598:differential form
1373:{\displaystyle n}
1317:{\displaystyle G}
1297:{\displaystyle G}
1274:{\displaystyle n}
1127:fundamental group
782:Cartan subalgebra
770:{\displaystyle K}
746:{\displaystyle K}
726:{\displaystyle T}
383:simple Lie groups
249:one can take its
87:discrete topology
77:realizes it as a
71:topological group
16:(Redirected from
5993:
5986:Fourier analysis
5962:
5944:
5926:
5908:
5902:
5896:
5890:
5884:
5878:
5872:
5866:
5860:
5854:
5848:
5842:
5836:
5830:
5824:
5823:Proposition 12.9
5818:
5812:
5806:
5800:
5798:
5779:
5773:
5770:
5759:
5753:
5747:
5741:
5735:
5729:
5723:
5717:
5711:
5705:
5699:
5590:
5588:
5587:
5582:
5570:
5568:
5567:
5562:
5560:
5559:
5539:
5537:
5536:
5531:
5529:
5528:
5508:
5506:
5505:
5500:
5498:
5497:
5481:
5479:
5478:
5473:
5461:
5459:
5458:
5453:
5451:
5450:
5433:
5431:
5430:
5425:
5413:
5411:
5410:
5405:
5403:
5402:
5383:
5381:
5380:
5375:
5373:
5372:
5356:
5354:
5353:
5348:
5336:
5334:
5333:
5328:
5326:
5325:
5309:
5307:
5306:
5301:
5289:
5287:
5286:
5281:
5269:
5267:
5266:
5261:
5249:
5247:
5246:
5241:
5239:
5238:
5220:
5218:
5217:
5212:
5198:
5196:
5195:
5190:
5188:
5187:
5168:
5166:
5165:
5160:
5158:
5157:
5141:
5139:
5138:
5133:
5121:
5119:
5118:
5113:
5101:
5099:
5098:
5093:
5081:
5079:
5078:
5073:
5058:
5056:
5055:
5050:
5048:
5046:
5045:
5044:
5001:
4985:
4984:
4983:
4928:
4912:
4904:
4903:
4891:
4876:
4874:
4873:
4868:
4856:
4854:
4853:
4848:
4846:
4834:
4832:
4831:
4826:
4814:
4812:
4811:
4806:
4786:
4784:
4783:
4778:
4741:
4739:
4738:
4733:
4721:
4719:
4718:
4713:
4701:
4699:
4698:
4693:
4669:
4667:
4666:
4661:
4643:
4641:
4640:
4635:
4614:
4612:
4611:
4606:
4530:
4528:
4527:
4522:
4517:
4516:
4495:
4494:
4458:
4457:
4427:
4426:
4408:
4404:
4403:
4396:
4395:
4366:
4365:
4341:
4323:
4321:
4320:
4315:
4310:
4308:
4291:
4259:
4254:
4250:
4249:
4242:
4241:
4212:
4211:
4187:
4168:
4166:
4165:
4160:
4145:
4143:
4142:
4137:
4125:
4123:
4122:
4117:
4105:
4103:
4102:
4097:
4078:
4076:
4075:
4070:
4065:
4064:
4057:
4056:
4027:
4026:
3929:
3927:
3926:
3921:
3919:
3903:
3901:
3900:
3895:
3883:
3881:
3880:
3875:
3863:
3861:
3860:
3855:
3844:
3833:
3832:
3811:
3795:
3793:
3792:
3787:
3736:
3721:
3719:
3718:
3713:
3711:
3697:
3685:
3683:
3682:
3677:
3662:, we define the
3657:
3655:
3654:
3649:
3591:
3589:
3588:
3583:
3581:
3580:
3559:
3557:
3556:
3551:
3539:
3537:
3536:
3531:
3518:integral element
3476:
3474:
3473:
3468:
3452:
3450:
3449:
3444:
3426:
3424:
3423:
3418:
3387:
3385:
3384:
3379:
3363:
3361:
3360:
3355:
3343:
3341:
3340:
3335:
3333:
3332:
3295:
3293:
3292:
3287:
3285:
3284:
3242:
3240:
3239:
3234:
3218:
3216:
3215:
3210:
3194:
3192:
3191:
3186:
3174:
3172:
3171:
3166:
3154:
3152:
3151:
3146:
3122:
3120:
3119:
3114:
3098:
3096:
3095:
3090:
3078:
3076:
3075:
3070:
3068:
3052:
3050:
3049:
3044:
2987:
2985:
2984:
2979:
2974:
2959:
2958:
2937:
2936:
2904:
2902:
2901:
2896:
2884:
2882:
2881:
2876:
2843:
2841:
2840:
2835:
2823:
2821:
2820:
2815:
2803:
2801:
2800:
2795:
2780:
2778:
2777:
2772:
2767:
2733:
2731:
2730:
2725:
2723:
2722:
2703:
2701:
2700:
2695:
2693:
2692:
2668:integral element
2661:
2659:
2658:
2653:
2641:
2639:
2638:
2633:
2631:
2616:
2614:
2613:
2608:
2590:
2560:
2558:
2557:
2552:
2540:
2538:
2537:
2532:
2520:
2518:
2517:
2512:
2500:
2498:
2497:
2492:
2473:
2471:
2470:
2465:
2450:
2448:
2447:
2442:
2437:
2433:
2426:
2425:
2407:
2406:
2373:
2371:
2370:
2365:
2353:
2351:
2350:
2345:
2343:
2342:
2322:
2320:
2319:
2314:
2302:
2300:
2299:
2294:
2292:
2291:
2266:
2264:
2263:
2258:
2256:
2255:
2242:
2240:
2239:
2234:
2219:
2217:
2216:
2211:
2209:
2208:
2181:
2180:
2155:
2153:
2152:
2147:
2132:
2130:
2129:
2124:
2119:
2118:
2102:
2101:
2076:
2074:
2073:
2068:
2046:
2044:
2043:
2038:
2036:
2035:
2019:
2017:
2016:
2011:
2009:
2001:
2000:
1984:
1982:
1981:
1976:
1957:
1955:
1954:
1949:
1944:
1943:
1938:
1926:
1880:
1878:
1877:
1872:
1857:is commutative,
1754:Cartan's theorem
1738:character theory
1719:
1717:
1716:
1711:
1699:
1697:
1696:
1691:
1671:
1670:
1650:
1648:
1647:
1642:
1630:
1628:
1627:
1622:
1567:modulus function
1504:Further examples
1498:
1496:
1495:
1490:
1488:
1487:
1469:
1467:
1466:
1461:
1459:
1458:
1439:
1437:
1436:
1431:
1429:
1428:
1410:
1408:
1407:
1402:
1400:
1399:
1379:
1377:
1376:
1371:
1355:
1353:
1352:
1347:
1323:
1321:
1320:
1315:
1303:
1301:
1300:
1295:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1228:
1226:
1225:
1220:
1196:
1194:
1193:
1188:
1164:
1162:
1161:
1156:
1088:
1086:
1085:
1080:
1078:
1077:
1061:
1059:
1058:
1053:
1024:
1022:
1021:
1016:
1014:
1013:
997:
995:
994:
989:
963:
961:
960:
955:
953:
952:
936:
934:
933:
928:
893:
891:
890:
885:
883:
882:
860:
858:
857:
852:
809:
807:
806:
801:
776:
774:
773:
768:
752:
750:
749:
744:
732:
730:
729:
724:
712:
710:
709:
704:
674:
672:
671:
666:
664:
663:
544:
542:
541:
536:
494:
492:
491:
486:
444:
442:
441:
436:
358:
356:
355:
350:
333:
332:
314:
313:
138:orthogonal group
102:Hausdorff spaces
21:
6001:
6000:
5996:
5995:
5994:
5992:
5991:
5990:
5966:
5965:
5960:
5942:
5917:
5912:
5911:
5903:
5899:
5891:
5887:
5883:Corollary 13.20
5879:
5875:
5867:
5863:
5855:
5851:
5843:
5839:
5831:
5827:
5819:
5815:
5807:
5803:
5780:
5776:
5760:
5756:
5748:
5744:
5736:
5732:
5724:
5720:
5712:
5708:
5700:
5696:
5691:
5641:
5617:unitarian trick
5613:
5597:
5576:
5573:
5572:
5555:
5551:
5549:
5546:
5545:
5524:
5520:
5518:
5515:
5514:
5493:
5489:
5487:
5484:
5483:
5467:
5464:
5463:
5446:
5442:
5440:
5437:
5436:
5419:
5416:
5415:
5398:
5394:
5392:
5389:
5388:
5368:
5364:
5362:
5359:
5358:
5342:
5339:
5338:
5321:
5317:
5315:
5312:
5311:
5295:
5292:
5291:
5275:
5272:
5271:
5255:
5252:
5251:
5234:
5230:
5228:
5225:
5224:
5206:
5203:
5202:
5183:
5179:
5177:
5174:
5173:
5153:
5149:
5147:
5144:
5143:
5127:
5124:
5123:
5107:
5104:
5103:
5087:
5084:
5083:
5067:
5064:
5063:
5019:
5015:
4991:
4986:
4946:
4942:
4918:
4913:
4911:
4899:
4895:
4887:
4885:
4882:
4881:
4862:
4859:
4858:
4842:
4840:
4837:
4836:
4820:
4817:
4816:
4800:
4797:
4796:
4772:
4769:
4768:
4727:
4724:
4723:
4707:
4704:
4703:
4687:
4684:
4683:
4676:
4649:
4646:
4645:
4623:
4620:
4619:
4549:
4546:
4545:
4503:
4499:
4469:
4465:
4435:
4431:
4416:
4412:
4398:
4397:
4385:
4381:
4379:
4373:
4372:
4367:
4358:
4354:
4347:
4346:
4342:
4337:
4335:
4332:
4331:
4292:
4260:
4258:
4244:
4243:
4231:
4227:
4225:
4219:
4218:
4213:
4204:
4200:
4193:
4192:
4188:
4183:
4181:
4178:
4177:
4154:
4151:
4150:
4131:
4128:
4127:
4111:
4108:
4107:
4091:
4088:
4087:
4059:
4058:
4046:
4042:
4040:
4034:
4033:
4028:
4019:
4015:
4008:
4007:
4005:
4002:
4001:
3991:
3915:
3913:
3910:
3909:
3889:
3886:
3885:
3869:
3866:
3865:
3840:
3825:
3821:
3807:
3805:
3802:
3801:
3732:
3730:
3727:
3726:
3707:
3693:
3691:
3688:
3687:
3671:
3668:
3667:
3619:
3616:
3615:
3612:
3606:
3576:
3575:
3573:
3570:
3569:
3545:
3542:
3541:
3525:
3522:
3521:
3462:
3459:
3458:
3432:
3429:
3428:
3397:
3394:
3393:
3373:
3370:
3369:
3349:
3346:
3345:
3328:
3327:
3325:
3322:
3321:
3280:
3279:
3271:
3268:
3267:
3228:
3225:
3224:
3204:
3201:
3200:
3180:
3177:
3176:
3160:
3157:
3156:
3140:
3137:
3136:
3108:
3105:
3104:
3084:
3081:
3080:
3064:
3062:
3059:
3058:
3038:
3035:
3034:
3003:
2997:
2970:
2948:
2944:
2929:
2925:
2917:
2914:
2913:
2890:
2887:
2886:
2849:
2846:
2845:
2829:
2826:
2825:
2809:
2806:
2805:
2786:
2783:
2782:
2763:
2739:
2736:
2735:
2715:
2711:
2709:
2706:
2705:
2688:
2684:
2682:
2679:
2678:
2647:
2644:
2643:
2627:
2625:
2622:
2621:
2586:
2569:
2566:
2565:
2546:
2543:
2542:
2526:
2523:
2522:
2506:
2503:
2502:
2483:
2480:
2479:
2459:
2456:
2455:
2415:
2411:
2402:
2401:
2394:
2390:
2382:
2379:
2378:
2359:
2356:
2355:
2338:
2334:
2332:
2329:
2328:
2308:
2305:
2304:
2287:
2283:
2275:
2272:
2271:
2251:
2250:
2248:
2245:
2244:
2228:
2225:
2224:
2192:
2188:
2176:
2172:
2164:
2161:
2160:
2141:
2138:
2137:
2114:
2113:
2097:
2093:
2085:
2082:
2081:
2056:
2053:
2052:
2031:
2030:
2028:
2025:
2024:
2005:
1996:
1992:
1990:
1987:
1986:
1970:
1967:
1966:
1939:
1934:
1933:
1922:
1893:
1890:
1889:
1866:
1863:
1862:
1851:
1806:
1726:
1705:
1702:
1701:
1666:
1662:
1660:
1657:
1656:
1636:
1633:
1632:
1616:
1613:
1612:
1559:
1553:
1545:discrete groups
1527:profinite group
1523:p-adic integers
1520:
1506:
1483:
1479:
1477:
1474:
1473:
1454:
1450:
1448:
1445:
1444:
1424:
1420:
1418:
1415:
1414:
1395:
1391:
1389:
1386:
1385:
1365:
1362:
1361:
1329:
1326:
1325:
1309:
1306:
1305:
1289:
1286:
1285:
1266:
1263:
1262:
1234:
1231:
1230:
1202:
1199:
1198:
1170:
1167:
1166:
1138:
1135:
1134:
1123:
1117:
1110:
1106:
1102:
1098:
1094:
1073:
1069:
1067:
1064:
1063:
1032:
1029:
1028:
1009:
1005:
1003:
1000:
999:
971:
968:
967:
948:
944:
942:
939:
938:
901:
898:
897:
872:
868:
866:
863:
862:
834:
831:
830:
789:
786:
785:
762:
759:
758:
738:
735:
734:
718:
715:
714:
680:
677:
676:
659:
655:
653:
650:
649:
630:
620:
597:Dynkin diagrams
581:
574:
567:
560:
553:
505:
502:
501:
455:
452:
451:
405:
402:
401:
392:
328:
324:
309:
305:
297:
294:
293:
280:
276:
258:
243:
219:
212:
205:
198:
191:
110:
35:
28:
23:
22:
15:
12:
11:
5:
5999:
5989:
5988:
5983:
5978:
5964:
5963:
5958:
5945:
5941:978-3319134666
5940:
5927:
5916:
5913:
5910:
5909:
5897:
5885:
5873:
5861:
5849:
5837:
5825:
5813:
5801:
5774:
5754:
5742:
5730:
5718:
5706:
5693:
5692:
5690:
5687:
5686:
5685:
5680:
5675:
5670:
5668:-compact group
5662:
5657:
5652:
5647:
5640:
5637:
5629:Harish-Chandra
5612:
5609:
5596:
5593:
5580:
5558:
5554:
5527:
5523:
5496:
5492:
5471:
5449:
5445:
5423:
5401:
5397:
5371:
5367:
5346:
5324:
5320:
5299:
5279:
5259:
5237:
5233:
5210:
5186:
5182:
5156:
5152:
5131:
5111:
5091:
5071:
5060:
5059:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5022:
5018:
5014:
5011:
5008:
5005:
5000:
4997:
4994:
4990:
4982:
4979:
4976:
4973:
4970:
4967:
4964:
4961:
4958:
4955:
4952:
4949:
4945:
4941:
4938:
4935:
4932:
4927:
4924:
4921:
4917:
4910:
4907:
4902:
4898:
4894:
4890:
4866:
4845:
4824:
4804:
4789:
4788:
4776:
4761:
4754:
4731:
4711:
4691:
4675:
4672:
4659:
4656:
4653:
4633:
4630:
4627:
4616:
4615:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4532:
4531:
4520:
4515:
4512:
4509:
4506:
4502:
4498:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4468:
4464:
4461:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4434:
4430:
4425:
4422:
4419:
4415:
4411:
4407:
4402:
4394:
4391:
4388:
4384:
4380:
4378:
4375:
4374:
4371:
4368:
4364:
4361:
4357:
4353:
4352:
4350:
4345:
4340:
4325:
4324:
4313:
4307:
4304:
4301:
4298:
4295:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4257:
4253:
4248:
4240:
4237:
4234:
4230:
4226:
4224:
4221:
4220:
4217:
4214:
4210:
4207:
4203:
4199:
4198:
4196:
4191:
4186:
4158:
4135:
4115:
4095:
4080:
4079:
4068:
4063:
4055:
4052:
4049:
4045:
4041:
4039:
4036:
4035:
4032:
4029:
4025:
4022:
4018:
4014:
4013:
4011:
3990:
3989:The SU(2) case
3987:
3918:
3893:
3873:
3853:
3850:
3847:
3843:
3839:
3836:
3831:
3828:
3824:
3820:
3817:
3814:
3810:
3798:
3797:
3785:
3782:
3779:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3735:
3710:
3706:
3703:
3700:
3696:
3675:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3608:Main article:
3605:
3602:
3579:
3549:
3529:
3510:
3509:
3506:
3503:
3500:
3466:
3442:
3439:
3436:
3416:
3413:
3410:
3407:
3404:
3401:
3377:
3353:
3331:
3283:
3278:
3275:
3232:
3208:
3184:
3164:
3144:
3112:
3088:
3067:
3042:
2996:
2990:
2989:
2988:
2977:
2973:
2969:
2966:
2962:
2957:
2954:
2951:
2947:
2943:
2940:
2935:
2932:
2928:
2924:
2921:
2894:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2833:
2813:
2793:
2790:
2770:
2766:
2762:
2759:
2755:
2752:
2749:
2746:
2743:
2721:
2718:
2714:
2691:
2687:
2651:
2630:
2618:
2617:
2606:
2603:
2600:
2597:
2593:
2589:
2585:
2582:
2579:
2576:
2573:
2550:
2530:
2510:
2490:
2487:
2463:
2452:
2451:
2440:
2436:
2432:
2429:
2424:
2421:
2418:
2414:
2410:
2405:
2400:
2397:
2393:
2389:
2386:
2363:
2354:. Rather, let
2341:
2337:
2312:
2290:
2286:
2282:
2279:
2254:
2232:
2221:
2220:
2207:
2204:
2201:
2198:
2195:
2191:
2187:
2184:
2179:
2175:
2171:
2168:
2145:
2134:
2133:
2122:
2117:
2112:
2109:
2105:
2100:
2096:
2092:
2089:
2066:
2063:
2060:
2034:
2008:
2004:
1999:
1995:
1974:
1959:
1958:
1947:
1942:
1937:
1932:
1929:
1925:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1870:
1850:
1844:
1805:
1802:
1775:inverse system
1725:
1722:
1709:
1689:
1686:
1683:
1680:
1677:
1674:
1669:
1665:
1640:
1620:
1575:positive reals
1552:
1549:
1516:
1505:
1502:
1486:
1482:
1457:
1453:
1427:
1423:
1398:
1394:
1369:
1345:
1342:
1339:
1336:
1333:
1313:
1293:
1270:
1250:
1247:
1244:
1241:
1238:
1218:
1215:
1212:
1209:
1206:
1186:
1183:
1180:
1177:
1174:
1154:
1151:
1148:
1145:
1142:
1116:
1113:
1112:
1111:
1108:
1104:
1100:
1096:
1092:
1089:
1076:
1072:
1051:
1048:
1045:
1042:
1039:
1036:
1025:
1012:
1008:
987:
984:
981:
978:
975:
964:
951:
947:
926:
923:
920:
917:
914:
911:
908:
905:
894:
881:
878:
875:
871:
850:
847:
844:
841:
838:
799:
796:
793:
766:
742:
722:
702:
699:
696:
693:
690:
687:
684:
662:
658:
619:
616:
579:
572:
565:
558:
551:
546:
545:
534:
531:
528:
524:
521:
518:
515:
512:
509:
495:
484:
481:
478:
474:
471:
468:
465:
462:
459:
445:
434:
431:
428:
424:
421:
418:
415:
412:
409:
388:
371:
370:
360:
359:
348:
345:
342:
339:
336:
331:
327:
323:
320:
317:
312:
308:
304:
301:
278:
274:
265:quotient group
256:
242:
241:Classification
239:
223:
222:
217:
210:
203:
196:
189:
180:
161:
134:
109:
106:
26:
9:
6:
4:
3:
2:
5998:
5987:
5984:
5982:
5979:
5977:
5974:
5973:
5971:
5961:
5959:3-11-015268-1
5955:
5951:
5946:
5943:
5937:
5933:
5928:
5924:
5919:
5918:
5906:
5901:
5894:
5889:
5882:
5877:
5870:
5865:
5858:
5853:
5846:
5841:
5834:
5829:
5822:
5817:
5810:
5805:
5797:
5793:
5789:
5785:
5778:
5772:
5768:
5764:
5758:
5751:
5746:
5739:
5734:
5727:
5722:
5715:
5710:
5703:
5698:
5694:
5684:
5681:
5679:
5676:
5674:
5671:
5669:
5667:
5663:
5661:
5658:
5656:
5653:
5651:
5650:Maximal torus
5648:
5646:
5643:
5642:
5636:
5634:
5630:
5626:
5622:
5618:
5608:
5606:
5602:
5592:
5578:
5556:
5543:
5525:
5512:
5494:
5469:
5447:
5421:
5399:
5385:
5369:
5344:
5322:
5297:
5277:
5257:
5235:
5222:
5208:
5184:
5172:Next, we let
5170:
5154:
5150:
5129:
5109:
5089:
5069:
5038:
5035:
5032:
5029:
5026:
5020:
5016:
5009:
4998:
4995:
4992:
4988:
4977:
4974:
4968:
4965:
4962:
4956:
4953:
4947:
4943:
4936:
4925:
4922:
4919:
4915:
4908:
4900:
4896:
4880:
4879:
4878:
4822:
4794:
4774:
4766:
4762:
4759:
4755:
4752:
4751:torus theorem
4748:
4747:
4746:
4743:
4729:
4709:
4689:
4681:
4671:
4657:
4654:
4651:
4631:
4628:
4625:
4602:
4599:
4596:
4593:
4587:
4584:
4581:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4544:
4543:
4542:
4540:
4535:
4518:
4513:
4510:
4507:
4504:
4500:
4496:
4491:
4485:
4482:
4479:
4473:
4470:
4466:
4462:
4459:
4454:
4448:
4445:
4442:
4436:
4432:
4428:
4423:
4420:
4417:
4413:
4409:
4405:
4400:
4392:
4389:
4386:
4382:
4376:
4369:
4362:
4359:
4355:
4348:
4343:
4330:
4329:
4328:
4311:
4302:
4296:
4293:
4285:
4279:
4276:
4273:
4264:
4261:
4255:
4251:
4246:
4238:
4235:
4232:
4228:
4222:
4215:
4208:
4205:
4201:
4194:
4189:
4176:
4175:
4174:
4172:
4156:
4147:
4133:
4113:
4093:
4085:
4066:
4061:
4053:
4050:
4047:
4043:
4037:
4030:
4023:
4020:
4016:
4009:
4000:
3999:
3998:
3996:
3986:
3984:
3980:
3976:
3972:
3968:
3967:Verma modules
3964:
3959:
3954:
3952:
3948:
3944:
3940:
3935:
3933:
3907:
3891:
3871:
3848:
3837:
3829:
3826:
3822:
3818:
3815:
3783:
3780:
3777:
3773:
3764:
3752:
3749:
3746:
3740:
3725:
3724:
3723:
3701:
3698:
3665:
3661:
3642:
3636:
3633:
3627:
3624:
3611:
3601:
3599:
3595:
3567:
3563:
3527:
3519:
3515:
3507:
3504:
3501:
3498:
3497:
3496:
3494:
3490:
3489:
3483:
3478:
3456:
3437:
3434:
3414:
3411:
3405:
3399:
3391:
3375:
3367:
3320:may not span
3319:
3315:
3311:
3307:
3303:
3299:
3276:
3273:
3265:
3261:
3257:
3253:
3248:
3246:
3222:
3206:
3198:
3162:
3142:
3134:
3130:
3126:
3102:
3056:
3027:
3020:
3019:eightfold way
3015:
3007:
3002:
2995:
2975:
2967:
2964:
2960:
2955:
2952:
2949:
2945:
2941:
2933:
2930:
2926:
2919:
2912:
2911:
2910:
2908:
2892:
2872:
2869:
2866:
2860:
2857:
2851:
2831:
2811:
2791:
2788:
2768:
2760:
2757:
2753:
2750:
2747:
2744:
2741:
2719:
2716:
2712:
2689:
2685:
2676:
2671:
2669:
2665:
2649:
2604:
2598:
2595:
2591:
2583:
2577:
2571:
2564:
2563:
2562:
2561:must satisfy
2548:
2528:
2508:
2488:
2485:
2477:
2461:
2438:
2434:
2430:
2427:
2422:
2419:
2416:
2412:
2408:
2398:
2395:
2391:
2387:
2377:
2376:
2375:
2339:
2335:
2326:
2310:
2288:
2284:
2277:
2268:
2230:
2202:
2196:
2193:
2189:
2185:
2177:
2173:
2166:
2159:
2158:
2157:
2143:
2120:
2110:
2107:
2103:
2098:
2094:
2090:
2087:
2080:
2079:
2078:
2064:
2061:
2058:
2050:
2021:
2002:
1997:
1993:
1972:
1964:
1961:Since, also,
1945:
1940:
1930:
1919:
1916:
1910:
1907:
1901:
1898:
1895:
1888:
1887:
1886:
1884:
1868:
1860:
1859:Schur's lemma
1856:
1849:
1843:
1841:
1837:
1834:
1833:maximal torus
1830:
1825:
1823:
1819:
1815:
1811:
1801:
1799:
1794:
1792:
1788:
1784:
1783:inverse limit
1780:
1776:
1772:
1767:
1763:
1759:
1755:
1750:
1747:
1743:
1742:maximal torus
1739:
1735:
1731:
1721:
1707:
1684:
1681:
1678:
1675:
1667:
1663:
1654:
1638:
1618:
1609:
1607:
1606:number theory
1603:
1599:
1595:
1594:Adolf Hurwitz
1590:
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1558:
1548:
1546:
1542:
1538:
1536:
1532:
1531:Galois groups
1528:
1524:
1519:
1515:
1511:
1501:
1499:
1484:
1480:
1470:
1455:
1451:
1441:
1425:
1421:
1396:
1392:
1381:
1367:
1359:
1340:
1334:
1331:
1311:
1291:
1282:
1268:
1245:
1239:
1236:
1213:
1207:
1204:
1181:
1175:
1149:
1143:
1140:
1132:
1128:
1122:
1090:
1074:
1070:
1046:
1043:
1037:
1034:
1026:
1010:
1006:
982:
976:
973:
965:
949:
945:
921:
918:
915:
912:
906:
903:
895:
879:
876:
873:
869:
845:
839:
836:
828:
827:
826:
823:
821:
817:
813:
797:
794:
791:
783:
778:
764:
756:
755:torus theorem
740:
720:
697:
691:
688:
685:
682:
660:
656:
647:
643:
639:
638:maximal torus
635:
629:
625:
624:Maximal torus
615:
613:
609:
606:. Indeed, if
605:
600:
598:
594:
590:
586:
582:
575:
568:
561:
554:
532:
529:
526:
522:
516:
510:
507:
500:
496:
482:
479:
476:
472:
466:
460:
457:
450:
446:
432:
429:
426:
422:
416:
410:
407:
400:
396:
395:
394:
391:
387:
384:
380:
375:
368:
365:
364:
363:
346:
337:
329:
325:
318:
310:
306:
299:
292:
291:
290:
288:
284:
273:
269:
266:
262:
255:
252:
248:
238:
236:
232:
228:
220:
213:
206:
199:
192:
185:
181:
178:
174:
170:
166:
165:unitary group
162:
159:
155:
151:
147:
143:
139:
135:
132:
129:
125:
122:
118:
117:
116:
114:
105:
103:
98:
96:
92:
91:group actions
88:
84:
83:finite groups
80:
76:
72:
68:
64:
60:
56:
48:
47:complex plane
44:
39:
33:
19:
5949:
5931:
5922:
5915:Bibliography
5900:
5888:
5876:
5871:Section 12.2
5864:
5852:
5847:Section 11.7
5840:
5835:Section 12.2
5828:
5816:
5804:
5787:
5783:
5777:
5771:
5766:
5757:
5752:Section 13.8
5745:
5733:
5721:
5709:
5697:
5665:
5614:
5598:
5541:
5510:
5386:
5200:
5171:
5061:
4790:
4744:
4680:Hermann Weyl
4677:
4617:
4536:
4533:
4326:
4171:in this case
4148:
4083:
4081:
3992:
3982:
3978:
3962:
3957:
3955:
3950:
3942:
3936:
3931:
3905:
3799:
3663:
3659:
3613:
3597:
3593:
3513:
3511:
3485:
3481:
3479:
3454:
3389:
3365:
3317:
3309:
3305:
3301:
3263:
3259:
3255:
3249:
3244:
3220:
3196:
3132:
3128:
3124:
3100:
3054:
3032:
2993:
2906:
2674:
2672:
2663:
2619:
2475:
2453:
2324:
2269:
2222:
2135:
2048:
2022:
1965:is compact,
1962:
1960:
1882:
1854:
1852:
1847:
1839:
1835:
1828:
1826:
1807:
1795:
1790:
1778:
1770:
1765:
1757:
1751:
1734:Hermann Weyl
1727:
1610:
1602:finite index
1591:
1578:
1571:homomorphism
1563:Haar measure
1560:
1551:Haar measure
1539:
1517:
1513:
1507:
1413:the compact
1382:
1357:
1356:consists of
1283:
1124:
824:
779:
754:
645:
641:
637:
633:
631:
611:
607:
601:
588:
584:
547:
389:
385:
378:
376:
372:
366:
361:
286:
282:
271:
267:
253:
246:
244:
224:
176:
168:
157:
149:
141:
130:
128:torus groups
123:
121:circle group
111:
99:
66:
62:
58:
52:
32:galaxy group
5790:: 737â755,
5763:Weil, André
5740:Section 7.7
5704:Section 1.2
5655:Root system
5623:there is a
4877:satisfies:
3314:root system
3256:root system
3033:We now let
812:root system
628:Root system
593:root system
259:, which is
233:and finite
63:topological
55:mathematics
5981:Lie groups
5970:Categories
5859:Chapter 12
5784:Math. Ann.
5728:Chapter 11
5689:References
3199:, we call
2999:See also:
1816:, and the
1583:unimodular
1555:See also:
1119:See also:
816:Weyl group
622:See also:
499:spin group
231:extensions
171:) and the
154:spin group
113:Lie groups
5905:Hall 2015
5893:Hall 2015
5881:Hall 2015
5869:Hall 2015
5857:Hall 2015
5845:Hall 2015
5833:Hall 2015
5821:Hall 2015
5809:Hall 2015
5750:Hall 2015
5738:Hall 2015
5726:Hall 2015
5702:Hall 2015
5579:λ
5557:λ
5553:Φ
5526:λ
5522:Φ
5495:λ
5491:Φ
5470:λ
5448:λ
5444:Φ
5422:λ
5400:λ
5396:Φ
5370:λ
5366:Φ
5345:λ
5323:λ
5319:Φ
5298:λ
5236:λ
5232:Φ
5209:λ
5185:λ
5181:Φ
5110:ρ
5042:⟩
5033:ρ
5030:⋅
5024:⟨
4996:∈
4989:∑
4981:⟩
4969:ρ
4963:λ
4957:⋅
4951:⟨
4923:∈
4916:∑
4865:Π
4823:λ
4803:Π
4597:−
4585:−
4576:−
4570:…
4561:−
4514:θ
4505:−
4492:θ
4483:−
4471:−
4463:⋯
4455:θ
4446:−
4424:θ
4393:θ
4387:−
4363:θ
4303:θ
4297:
4286:θ
4265:
4239:θ
4233:−
4209:θ
4054:θ
4048:−
4024:θ
3827:−
3781:∈
3759:Π
3753:
3722:given by
3705:→
3674:Π
3664:character
3637:
3631:→
3622:Π
3548:Σ
3528:λ
3465:Δ
3441:Δ
3438:∈
3435:α
3412:≥
3406:α
3400:λ
3376:λ
3352:Δ
3277:⊂
3231:Σ
3207:λ
3183:Σ
3163:λ
3143:λ
3111:Σ
3087:Σ
3041:Σ
2968:∈
2956:θ
2934:θ
2920:ρ
2873:θ
2861:θ
2852:λ
2832:λ
2761:∈
2758:θ
2751:θ
2720:θ
2650:λ
2602:Γ
2599:∈
2584:∈
2572:λ
2549:λ
2529:ρ
2509:λ
2489:π
2420:π
2409:∣
2399:∈
2385:Γ
2362:Γ
2311:λ
2281:↦
2231:λ
2197:λ
2167:ρ
2144:ρ
2111:∈
2062:∈
2003:⊂
1973:ρ
1941:∗
1905:→
1896:ρ
1869:ρ
1335:
1240:
1208:
1176:
1144:
1038:
977:
907:
877:−
840:
795:⊂
692:
530:≥
511:
480:≥
461:
430:≥
411:
344:→
326:π
322:→
316:→
303:→
261:connected
85:with the
5811:Part III
5765:(1940),
5673:Protorus
5639:See also
5607:theory.
5201:even if
5062:for all
3908:. Thus,
3864:for all
3562:integral
3427:for all
3390:dominant
1510:manifold
126:and the
75:topology
5595:Duality
5102:. Here
367:Theorem
144:), the
59:compact
5956:
5938:
5511:proper
5414:âwith
3488:weight
3455:higher
3221:weight
3057:(over
2804:where
2620:where
2454:where
1853:Since
1831:and a
1812:, the
1781:as an
1229:, and
1107:, or E
814:and a
576:, and
263:. The
235:covers
214:, and
73:whose
43:circle
5542:basis
3958:after
3750:trace
3017:The "
2327:into
1764:Ï of
1440:group
156:Spin(
69:is a
67:group
5954:ISBN
5936:ISBN
4763:The
4756:The
4749:The
3884:and
3364:for
3258:for
1471:and
1035:Spin
904:Spin
626:and
508:Spin
497:The
447:The
397:The
225:The
163:the
136:the
119:the
93:and
57:, a
41:The
5792:doi
5004:det
4931:det
4857:of
4294:sin
4262:sin
3904:in
3666:of
3614:If
3392:if
3388:is
3223:of
3195:to
3099:to
2243:on
2077:as
1881:of
1838:in
1789:of
1611:If
1573:to
1521:of
1500:.)
1103:, E
1099:, E
1095:, F
644:of
175:SU(
148:SO(
53:In
5972::
5788:97
5786:,
5223:,
4146:.
3985:.
3934:.
3634:GL
3219:a
2541:,
2462:Id
2431:Id
2267:.
2020:.
1842:.
1824:.
1732:.
1720:.
1608:.
1547:.
1380:.
1332:SU
1237:Sp
1205:SO
1197:,
1165:,
1141:SU
974:Sp
837:SU
689:SU
599:.
569:,
562:,
555:,
458:SU
408:Sp
347:1.
207:,
200:,
193:,
186::
179:),
167:U(
160:),
140:O(
104:.
97:.
65:)
5799:.
5794::
5666:p
5278:K
5258:T
5155:2
5151:L
5130:i
5090:T
5070:H
5039:H
5036:,
5027:w
5021:i
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150:n
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133:,
131:T
124:T
61:(
34:.
20:)
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