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Is the Luna stratification intrinsic? (English) Zbl 1145.14047
Let \(V\) be a finite dimensional \(k\)-vector space where \(k\) is an algebraically closed field of characteristic zero, and \(G\to GL(V)\) a representation of a reductive linear algebraic group. The categorical quotient, \(X=V//G\) has a stratification, defined by D. Luna, where the strata are given as follows. Let \(\pi: V\to X\) be the quotient map. For each point \(p\in X\), the fibre \(\pi^{-1}(p)\) has a unique closed orbit. Let \(v_p\) be a point on such an orbit, and let \(H\) be the isotropy group of \(v_p\). Denote by \((H)\) the conjugacy class of \(H\). Then a stratum in \(X\) is the set \(X^{(H)}=\{p\in X\mid Stab(v_p)\in(H)\}\). There are finitely many strata, and they are locally closed. In this article, the authors consider the following questions: (i) Does every automorphism of \(X\) map each stratum into another stratum? (Is the stratification intrinsic?) (2) Does every automorphism of \(X\) preserve each stratum? (Are the individual strata intrinsic?)
In general, the answer to both questions is negative. For example, there are many cases where the quotient \(X\) is affine space, in which case the group of automorphisms is transitive. However, under some conditions, there are positive results to these questions. For example, as described in the present article, if \(G\) is finite and contains no pseudo-groups, the answer to the first question is positive, and under some other assumptions, one can show that the strata are intrinsic. For infinite reductive groups, the authors give conditions under which the Luna stratification is intrinsic. Also for a certain family of representations, they show that even the strata are intrinsic. More precisely, it is shown that if \(G\to GL(W)\) is a finite-dimensional representation of a reductive algebraic group \(G\) and \(V=W^r\), where any of the following three conditions hold : (i) \(r\geq 2\dim (W)\); or (ii) \(G\) preserves a nondegenerate quadratic form on \(W\) and \(r\geq\dim(W)+1\); or (iii) \(W={\mathbf g}\) is the adjoint representation of \(G\) and \(r\geq3\), then the Luna stratification of \(X=V//G\) is intrinsic. As for the second question, if \(V=M_n^r\) is the space of \(r\)-tuples of \(n\times n\) matrices, with \(r\geq 3\), and \(G=GL_n\) acts on \(V\) by simultaneous conjugation, then every Luna stratum of the categorical quotient \(V//GL_n\) is intrinsic.

MSC:
14R20 Group actions on affine varieties
14L40 Other algebraic groups (geometric aspects)
14B05 Singularities in algebraic geometry
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