Canonical Bases and the Conjugating Representation of a Semisimple Group

We fix an algebraically closed field k of characteristic zero. Let G be a reductive affine algebraic group over k and let V be an affine G-variety over k. We denote by A(G) and A(V ) the k-algebras of regular functions on G and V respectively. The action of G on V gives rise to a rational representation of G on A(V ). A natural question is to investigate whether the algebra A(V ) is a free module over its subalgebra A(V )G of invariant elements. The case where V is a k-vector space on which G acts linearly has been investigated by Chevalley [Ch, Bo], Kostant [Ko], Popov [Po], Schwarz [Sc], and Littelmann [Li]. In the general case, only examples have been studied, for instance by Richardson [Ri1, Ri2] or Schwarz and Wehlau [SW]. We will investigate the case where the variety V is the group G, acting on itself by inner automorphisms. Then the subalgebra of invariant elements C(G) = A(G)G is the set of regular class functions. We assume in the remainder of the paper thatG is semisimple and simply connected. Richardson proved in [Ri1] that the following result holds under these assumptions.