Isomorphisms between orbit spaces (Q1061188)

The purpose of this note is to suggest a certain modification of the Chevalley-Luna-Richardson theorem on restriction of invariants [see \textit{D. Luna} and \textit{R. W. Richardson}, Duke Math. J. 46, 487--496 (1979; Zbl 0444.14010)]. Namely, let \(V\) be a rational \(G\)-module, where \(G\) is a reductive algebraic group over an algebraically closed field of characteristic 0. Let \(X\subset V\) be a conical \(G\)-invariant irreducible normal closed algebraic subvariety of \(V\). Let \(PV\) be the projective space canonically associated to \(V\) (the space of lines through the origin in \(V\)) and \(PX\) be the image of \(X\) under the natural projection \(V\setminus 0\to PV.\) Assume that the \(G\)-stabilizer of the ``general'' closed orbit of \(G\) in \(X\) is trivial. Then the following theorem is proved: Assume that \(PX\) is smooth. Then: (1) there exists a stabilizer of general position \(H'\) for the action of \(G\) on \(PX\); (2) assume that not all of the elements of \(H'\) act trivially on \(PX\); then there exists such an irreducible component \(PY\) of the variety \((PX)^{H'}\) that the restriction of functions induces the isomorphism \(Y/\{g\in Z_ G(H')\mid gY=Y\}=X/G\).