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


22E46 Semisimple Lie groups and their representations
17B20 Simple, semisimple, reductive (super)algebras
Full Text: DOI


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