Stably Trivial Equivariant Algebraic Vector Bundles

Let G be a reductive algebraic group over e, let F be a G-module, and let B be an affine G-variety, i.e., an affine variety with an algebraic action of G. Then B x F is in a natural way a G-vector bundle over B, which we denote by F. (All vector bundles here are algebraic.) A G-vector bundle over B is called trivial if it is isomorphic to F for some G-module F . From the endomorphism ring R of the G-vector bundle S, we construct G-vector bundles over B . The bundles constructed this way have the property that when added to S they are isomorphic to F EB S for a fixed G-module F. They are called stably trivial. The set of isomorphism classes of G-vector bundles over B which satisfy this condition is denoted by VEC(B, F; S). For such a bundle E we define an invariant p(E) which lies in a quotient of R. This invariant allows us to distinguish non-isomorphic G-vector bundles. When B is a Gmodule, a G-vector bundle over B defines an action of G on affine space. We give criteria which in certain cases allow us to distinguish the underlying actions. The construction and invariants are applied to the following two problems: