Étale Slices for Algebraic Transformation Groups in Characteristic p

Abstract

A.M.S. {1980) subject classification: 20G 15.We summarize some standard results on G-invariants for morphic actions of G on an affine variety X. For more details on some of these results see [5,10,11,13]. These references generally deal with the case in which G is reductive (and hence connected). However, the extension to the case of non-connected G follows easily from the results of [17, pp. 57-60]. The references [10,11] deal with the characteristic-zero case, but many of the foundational results in these references carry over to characteristic p because of the affirmative solution of the Mumford Conjecture by Haboush. See the appendix to [11] for a discussion of this point.2.1. The quotient variety X/G. Let G act morphically on an affine variety X. Then the algebra of invariants k[X~\ G is a finitely generated /c-algebra. We denote by X/G the affine variety Spm^A'] 0 ) and by n x : X -> X/G the morphism of varieties corresponding to the inclusion homomorphism k[X] G -> /c [X]; when reference to G is necessary we write n x G instead of n x .The morphism n x has the following properties.2.1.1. The morphism n x is a surjective morphism.2.1.2. If V x and V 2 are disjoint, G-stable closed subsets of X, then there exists an invariant fe k[X] G such that / = 0 on V x and / = 1 on V 2 . (a)If c e X/G, then the fibre n x~l (£,) contains a unique closed orbit. We denote by T(£) the unique closed orbit in n x~l {Q.(b) n x determines a bijective mapping from the set of closed G-orbits on X to X/G.2.1.4. If V is a G-stable closed subset of X, then n x (V) is a closed subset of X/G. 2.1.5. The topology on X/G is the quotient topology. That is, a subset U of X/G is open if and only if n x~l (U) is open in X.

Localization of modules for a semisimple Lie algebra in prime characteristic (with an Appendix by R. Bezrukavnikov and S. Riche: Computation for sl(3))

“…Thus it suffices to show that the global functions on the nilpotent variety N ⊂ g map isomorphically to the ring of global functions on N ∼ = n × B G. Moreover, theétale slice theorem of [BaRi] shows that for very good p there exists a G-equivariant isomorphism between N and the subscheme U ⊂ G defined by the G-invariant polynomials on G vanishing at the unit element; cf. [BaRi,9.3]. Thus the task is reduced to showing that the ring of regular functions on U maps isomorphically to the ring of global functions on N × B G. This follows once we know that U is reduced and normal and the Springer map N × B G → U is birational.…”

Nilpotent commuting varieties of reductive Lie algebras

“…In fact, we need the G-equivariant isomorphism between U and N introduced in [3]. This isomorphism, call it η, is defined in loc.…”

Nilpotent orbits over ground fields of good characteristic

“…The fact that Λ satisfies condition (2) of the statement of the theorem follows from Corollary 9.3.4 of [BR85]; condition (3) follows from Theorem 6.2 in loc. cit.…”