×

zbMATH — the first resource for mathematics

Représentations exceptionnelles des groupes semi-simples. (French) Zbl 0588.22010
In this paper the rational representations \(\phi\) : \(G\to GL(V)\) are studied, where V is a finite dimensional vector space and G is a semisimple group over \({\mathbb{C}}\) satisfying the following condition: the algebra \({\mathbb{C}}[V]^ U\) of U-invariant functions on V, where U is a maximal unipotent subgroup of G, is free. In this case V and the corresponding representation are called exceptional.
In the first part of the paper the closure of G-orbits in V are characterized. The second part of the paper is devoted to the classification of exceptional representations of simple groups. In the third part it is established that all singularities of the variety \(\overline{Gx}\), \(x\in V\), lie in its boundary.
[The results of this paper were announced in C. R. Acad. Sci., Paris, Sér. I 296, 5-6 (1983; Zbl 0538.14007)].
Reviewer: G.A.Soifer

MSC:
22E46 Semisimple Lie groups and their representations
14L24 Geometric invariant theory
20G05 Representation theory for linear algebraic groups
14J17 Singularities of surfaces or higher-dimensional varieties
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] G. SCHWARZ , Representations of Simple Lie Groups with Regular Rings of Invariants (Invent. Math., vol. 49, 1978 , p. 167-191). MR 80m:14032 | Zbl 0391.20032 · Zbl 0391.20032 · doi:10.1007/BF01403085 · eudml:142600
[2] G. SCHWARZ , Representations of Simple Lie Groups with a Free Module of Covariants (invent. Math., vol. 50, 1978 , p. 1-12). MR 80c:14008 | Zbl 0391.20033 · Zbl 0391.20033 · doi:10.1007/BF01406465 · eudml:142603
[3] V. KAČ , Some Remarks on Nilpotent Orbits (J. Alg., vol. 64, 1980 , p. 190-213). MR 81i:17005 | Zbl 0431.17007 · Zbl 0431.17007 · doi:10.1016/0021-8693(80)90141-6
[4] H. KRAFT et C. PROCESI , Closures of Conjugacy Classes of Matrices are Normal (Invent. Math., vol. 53, 1979 , p. 227-247). MR 80m:14037 | Zbl 0434.14026 · Zbl 0434.14026 · doi:10.1007/BF01389764 · eudml:142663
[5] M. BRION , Sur certaines représentations des groupes semi-simples (C.R. Acad. Sc., t. 296, série I, 1983 , p. 5-6). MR 84b:14027 | Zbl 0538.14007 · Zbl 0538.14007
[6] M. BRION , Invariants d’un sous-groupe unipotent maximal d’un groupe semi-simple (Ann. Inst. Fourier, t. 33, fasc. I, 1983 , p. 1-27). Numdam | MR 85a:14031 | Zbl 0475.14038 · Zbl 0475.14038 · doi:10.5802/aif.902 · numdam:AIF_1983__33_1_1_0 · eudml:74571
[7] R. FOSSUM et B. IVERSEN , On Picard Groups of Some Algebraic Fiber Spaces (J. Pure Appl. Algebra, vol. 3, 1973 , p. 269-280). MR 50 #9864 | Zbl 0277.14005 · Zbl 0277.14005 · doi:10.1016/0022-4049(73)90014-5
[8] G. KEMPF , Toroidal Embeddings (Springer L.N., n^\circ 339). MR 49 #299 | Zbl 0271.14017 · Zbl 0271.14017
[9] N. BOURBAKI , Groupes et algèbres de Lie , chap. VII et VIII, Hermann. · Zbl 0329.17002
[10] D. E. LITTLEWOOD , The Theory of Group Characters , Oxford University Press. · Zbl 0011.25001
[11] I. G. MACDONALD , Symmetric Functions and Hall Polynomials , Oxford University Press. · Zbl 0899.05068
[12] D. E. LITTLEWOOD , Products and Plethysms of Characters with Orthogonal, Symplectic and Symmetric Groups (Can. J. Math., vol. 10, 1958 , p. 17-32). MR 20 #1715 | Zbl 0079.03604 · Zbl 0079.03604 · doi:10.4153/CJM-1958-002-7
[13] H. WEYL , Cesammelte Abhandlungen , Band III.
[14] R. C. KING , Spinor representations (Lecture Notes in Physics, n^\circ 50) ; Group Theoretical Methods in Physics, Springer-Verlag. MR 58 #1043 | Zbl 0369.22017 · Zbl 0369.22017
[15] D. E. LITTLEWOOD , On the Concomitants of Spin Tensors (Proc. London Math. Soc., (2), vol. 49, 1947 , p. 307-327). MR 9,76c | Zbl 0033.00902 · Zbl 0033.00902 · doi:10.1112/plms/s2-49.4.307
[16] M. KRÄMER , Eine Klassifikation bestimmter Untergruppen kompakter Liegruppen (Comm. in Algebra, vol. 3, 1975 , p. 691-737). MR 51 #13140 | Zbl 0309.22013 · Zbl 0309.22013 · doi:10.1080/00927877508822068
[17] W. LICHTENSTEIN , A System of Quadrics Describing the Orbit of the Highest Weight Vector (Proc. A.M.S., vol. 84, n^\circ 4, avril 1982 ). MR 83h:14046 | Zbl 0501.22017 · Zbl 0501.22017 · doi:10.2307/2044044
[18] E. VINBERG et V. POPOV , On a Class of Quasihomogeneous Varieties (Math. U.S.S.R. Izv., vol. 6, 1972 , p. 743-748). MR 47 #1815 | Zbl 0255.14016 · Zbl 0255.14016 · doi:10.1070/IM1972v006n04ABEH001898
[19] W. BORHO et H. KRAFT , Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen (Comm. Math. Helv., vol. 54, 1979 , p. 61-104). MR 82m:14027 | Zbl 0395.14013 · Zbl 0395.14013 · doi:10.1007/BF02566256 · eudml:139768
[20] S. J. HARIS , Some Irreducible Representations of Exceptional Algebraic Groups (Amer. J. Maths., vol. 93, 1971 , p. 75-106). MR 43 #4829 | Zbl 0215.39603 · Zbl 0215.39603 · doi:10.2307/2373449
[21] J. G. MARS , Les nombres de Tamagawa de certains groupes exceptionnels (Bull. Soc. Math. Fr., t. 94, 1966 , p. 97-140). Numdam | MR 35 #4227 | Zbl 0146.04601 · Zbl 0146.04601 · numdam:BSMF_1966__94__97_0 · eudml:87091
[22] J. I. IGUSA , A Classification of Spinors up to Dimension Twelve (Amer. J. Math., vol. 92, 1970 , p. 997-1028). MR 43 #3291 | Zbl 0217.36203 · Zbl 0217.36203 · doi:10.2307/2373406
[23] G. KEMPF , Images of Homogeneous Vector Bundles and Varieties of Complexes (Bull. Amer. Math. Soc., vol. 81, 1975 , n^\circ 5, p. 900-901). Article | MR 52 #5689 | Zbl 0322.14020 · Zbl 0322.14020 · doi:10.1090/S0002-9904-1975-13878-X · minidml.mathdoc.fr
[24] C. DE CONCINI et E. STRICKLAND , On the Variety of Complexes (Adv. in Maths, vol. 41, 1981 , n^\circ 1, p. 57-77). MR 82m:14032 | Zbl 0471.14026 · Zbl 0471.14026 · doi:10.1016/S0001-8708(81)80004-7
[25] Th. VUST , Sur la théorie des invariants des groupes classiques (Ann. Inst. Fourier, vol. 26, n^\circ 1, 1976 , p. 1-31). Numdam | MR 53 #8082 | Zbl 0314.20035 · Zbl 0314.20035 · doi:10.5802/aif.598 · numdam:AIF_1976__26_1_1_0 · eudml:74265
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.