Étale slices for representation varieties in characteristic p

Abstract

Let R( F, G) be the variety of representations of a finitely generated group F into a connected reductive algebraic group G, and let C( F, G) be the variety of closed conjugacy classes of representations. We examine the question of whether an étale slice for the conjugation action of G exists through a representation ρR( F, G) when the ground field k has characteristic p > 0. We show that an étale slice through ρ may exist for the action of an enlarged group Ĝ, even when there is no étale slice for the G-action. As an application, we generalise a result known to hold in characteristic zero, which expresses the tangent space to C( F, G) at the conjugacy class of a suitable representation ρ as a subspace of the 1-cohomology H 1 ( F, %plane1D;524;(ρ)) of an F-module %plane1D;524;(ρ). A similar result holds in characteristic p, but with H 1 ( F%plane1D;524;(ρ)) replaced by a quotient of H 1 ( F%plane1D;524;(ρ)).