Linearizing Flat Families of Reductive Group Representations

Abstract

Let G be a reductive group over C. Let X → S be a G-morphism of Gvarieties where G acts trivially on S. Then for p ∈ S the fiber X(p) over p is a G-variety, so we can view X as an S-parametrized family (X(p))_p∈S of G-varieties. We ask here, under various linearity assumptions on the individual X(p)’s, what kind of global linearity properties X must have over S. For example, assuming that X is flat over S and that each X(p) is isomorphic to a G-representation space, one would like to conclude that X is isomorphic to a G-vector bundle over S. The results announced here imply that this is “stably” true, i.e. X × V is a G-vector bundle over S for some G-representation space V.