Lifting automorphisms of generalized adjoint quotients

Abstract

Let G be a reductive algebraic group over an algebraically closed field K of characteristic zero. Let π : g r → X = g r / /G be the categorical quotient where g is the adjoint representation of G and r is a suitably large integer (in general r ≥ 5, but for many cases r ≥ 3 or even r ≥ 2 suffices). We show that every automorphism ϕ of X lifts to a map Φ : g r → g r commuting with π. As an application we consider the action of ϕ on the Luna stratification of X.

Quotients, automorphisms and differential operators

“…Thus, ϕ sends the stratum with conjugacy class (H) to the stratum with conjugacy class (σ(H)), that is, ϕ permutes the strata of Z according to the action of σ on conjugacy classes. Kuttler [9] has a similar result for ϕ algebraic.…”

“…which is exact on V pr . The key idea, following [9], is to show that Γ(V pr , π # ϕ * F ) = Γ(V, π # ϕ * F ) is a free O(V )-module. Now we take cohomology over V pr to get an exact sequence…”

“…where the last module is zero since M has depth 3 with respect to the ideal of V 0 . The map H 1 (V pr , π # ϕ * E) → H 1 (V pr , π # ϕ * F) is a surjective morphism of finitely generated O(V )modules (see [9,Remark 13]). Since π # ϕ * F contains π # ϕ * E as a direct summand, then localizing at points of V and applying Nakayama's lemma, we see that H 1 (V pr , π # ϕ * E) → H 1 (V pr , π # ϕ * F) has to be injective.…”

“…Then S is not in the closure of any stratum except the principal one. From [9], we borrow the following lemma. (1) If S is subprincipal, then it has codimension 4.…”

“…We first consider the case where codim S = 5 and H has rank 1. We use some of the same techniques as [9]. Let v ∈ V such that π(v) ∈ S. Then there is a 1-parameter subgroup λ : C * → G such that v 0 := lim t→0 λ(t) · v exists where Gv 0 is closed and G v0 is conjugate to H. Applying an element of G we may assume that G v0 = H. Since λ fixes v 0 , λ is a 1-parameter subgroup of H where H 0 = C * .…”