Support varieties for modules over Chevalley groups and classical Lie algebras

Abstract

Let G G be a connected reductive algebraic group over an algebraically closed field of characteristic p > 0 p>0 , G 1 G_{1} be the first Frobenius kernel, and G ( F p ) G({\mathbb F}_{p}) be the corresponding finite Chevalley group. Let M M be a rational G G -module. In this paper we relate the support variety of M M over the first Frobenius kernel with the support variety of M M over the group algebra k G ( F p ) kG({\mathbb F}_{p}) . This provides an answer to a question of Parshall. Applications of our new techniques are presented, which allow us to extend results of Alperin-Mason and Janiszczak-Jantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.