K-THEORY OF AFFINE ACTIONS

For a Lie group G and a vector bundle E we study those actions of the Lie group TG on E for which the action map TG x E -> E is a morphism of vector bundles, and call those affine actions. We prove that the category Vect(TG)(aff)(X) of such actions over a fixed G-manifold X is equivalent to a certain slice category g(X)\Vect(G )(X). We show that there is a monadic adjunction relating Vect(TG)(aff)(X) to Vect(G) (X), and the right adjoint of this adjunction induces an isomorphism of Grothendieck groups K-TG(aff) (X) congruent to K O-G (X) . Complexification produces analogous results involving T(C)G and K-G(X).