Is the Luna stratification intrinsic?

Abstract

Let G → GL(V ) will be a finite-dimensional linear representation of a reductive linear algebraic group G on a finite-dimensional vector space V , defined over an algebraically closed field of characteristic zero. The categorical quotient V // G carries a natural stratification, due to D. Luna. This paper addresses the following questions:(i) Is the Luna stratification of X intrinsic? That is, does every automorphism of V // G map each stratum to another stratum?(ii) Are the individual Luna strata in X intrinsic? That is, does every automorphism of V // G maps each stratum to itself?In general, the Luna stratification is not intrinsic. Nevertheless, we give positive answers to questions (i) and (ii) for large classes of interesting representations.

Torsors over Luna strata

“…Following [6] we say that a Luna stratum X H is singular along its boundary if the singular locus of the closure X H is precisely the complement X H \ X H . This property is known for the principal stratum on the quotient variety of the space of representations of a quiver except for some low dimensional anomalies [7,Theorem 7].…”

“…This property is known for the principal stratum on the quotient variety of the space of representations of a quiver except for some low dimensional anomalies [7, Theorem 7]. Moreover, there are many modules where every Luna stratum is singular along its boundary, see [6,Theorem 1.2]. In particular, this is the case for the module V ⊕r , where V is any rational G-module and r ≥ 2 dim(V ), or where V is the adjoint module and r ≥ 3.…”

“…It turns out that in many cases a Luna stratum is singular along its boundary, i.e. the stratum coincides with the smooth locus of its closure in X, see [15,Theorem 8.1], [7,Theorem 7] and [6,Theorem 1.2]. In [6], J. Kuttler and Z. Reichstein use this property to prove that quite often the Luna stratification is intrinsic.…”

“…the stratum coincides with the smooth locus of its closure in X, see [15,Theorem 8.1], [7,Theorem 7] and [6,Theorem 1.2]. In [6], J. Kuttler and Z. Reichstein use this property to prove that quite often the Luna stratification is intrinsic. The latter means that every automorphism of X as of an abstract affine variety preserves the stratification possibly permuting the strata.…”

“…In particular, we prove in Proposition 1 that if H is the isotropy group of a point with a closed G-orbit in a module V , then the Luna stratum corresponding to H and to the G-module V ⊕k with k ≥ 2 is admissible. The idea to obtain strata with good properties by replacing a module V by V ⊕k is taken from [6]. In Section 4 we formulate the main result (Theorem 2) and obtain some corollaries.…”