Polar representations
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.
scientific article; zbMATH DE number 3987583