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Polar representations. (English) Zbl 0611.22009

Let G be a complex affine reductive algebraic group acting on a complex vector space V by a rational representation. Let \(v\in V\) be a vector with a closed G-orbit. Define a subspace \(c_ v=\{x\in V|\) \({\mathfrak g}\cdot x\subset {\mathfrak g}\cdot v\}\), \({\mathfrak g}\) the Lie algebra of V. If dim \(c_ v=\dim {\mathbb{C}}[V]^ G\) the action is called polar and \(c=c_ v\) a Cartan subspace. Assume this. The authors show: (1) Cartan subspaces are G-conjugate; (2) all closed orbits intersect c; (3) \(W=N_ G(c)/Z_ G(c)\) is finite and V/G\(\approx c/W\); (4) \({\mathbb{C}}[V]^ G\approx {\mathbb{C}}[c]^ W\); (5) W is a pseudo-reflection group.
A list of all polar representations for G connected, simple is given.
Reviewer: W.Rossmann

MSC:

22E46 Semisimple Lie groups and their representations
17B20 Simple, semisimple, reductive (super)algebras
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[2] Kac, V. G., On the orbit classification for a linear algebraic group, Uspekhi Mat. Nauk, 30, 173-174 (1975), (in Russian) · Zbl 0391.20035
[3] Kac, V. G., Some remarks on nilpotent orbits, J. Algebra, 64, 190-213 (1980) · Zbl 0431.17007
[4] Kac, V. G.; Popov, V. L.; Vinberg, E. B., Sur les groupes linéares algebrique dont l’algèbre des invariants est libre, C. R. Acad. Sci. Paris, 283, 875-878 (1976) · Zbl 0343.20023
[5] Gatti, V.; Viniberghi, E., Spinors of 13-dimensional space, Adv. in Math., 30, 137-155 (1978) · Zbl 0429.20043
[6] Kempf, G.; Ness, L., The length of vectors in representation spaces, (Algebraic Geometry. Algebraic Geometry, Lecture Notes in Mathematics No. 732 (1978), Springer-Verlag: Springer-Verlag Copenhagen) · Zbl 0407.22012
[7] Kostant, B., Lie group representations on polynomial rings, Amer. J. Math., 85, 327-402 (1963) · Zbl 0124.26802
[8] Kostant, B.; Rallis, S., Orbits and representations associated with symmetric spaces, Amer. J. Math., 93, 753-809 (1971) · Zbl 0224.22013
[9] Luna, D., Adhérences d’orbites et invariants, Invent. Math., 29, 231-238 (1975) · Zbl 0315.14018
[10] Motzkin, T. S.; Taussky-Todd, O., Pairs of matrices with property \(L\), TAMS, 387-401 (Nov. 1955)
[11] Mumford, D., Geometric Invariant Theory (1965), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0147.39304
[12] Panjushev, D. I., On orbit spaces of finite and connected linear groups, Izvestiya, 46, 95-99 (1982), (in Russian)
[13] Popov, V. L., Representations with a free module of covariants, Functional Anal. Appl., 10, 242-244 (1976) · Zbl 0365.20053
[14] Schwartz, G. W., Representation of simple Lie groups with a free module of covariants, Invent. Math., 50, 1-12 (1978) · Zbl 0391.20033
[15] Shafarevich, I. R., Basic Algebraic Geometry (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0362.14001
[16] Shepard, G. C.; Todd, A. J., Finite unitary reflection groups, Canad. J. Math., 6, 274-304 (1954) · Zbl 0055.14305
[17] Springer, T. A., Regular elements of finite reflection groups, Invent. Math., 25, 159-198 (1974) · Zbl 0287.20043
[18] Vinberg, E. B., The Weyl group of a graded Lie algebra, Math. USSR-Izv., 10, 463-495 (1976) · Zbl 0371.20041
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