A Harish-Chandra Homomorphism for Reductive Group Actions

Consider a semisimple complex Lie algebra g and its universal enveloping algebra U(g). In order to study unitary representations of semisimple Lie groups, Harish-Chandra ([HC1] Part III) established an isomorphism between the center Z(g) of U(g) and the algebra of invariant polynomials C[t] . Here, t ⊆ g is a Cartan subspace and W is the Weyl group of g. This is one of the most basic results in representation theory. Later on ([HC2] Thm. 1), he found a similar isomorphism for a symmetric space X = G/K. Here, instead of Z(g), he considered the algebra D(X) of invariant differential operators on X and showed, that it is isomorphic to the ring of invariants of the little Weyl group WX attached to X. This isomorphism is very important for analyzing the action of G on various function spaces on X. Actually, it is a generalization of the former result if one considers the natural G×G-action on X = G. The proofs of these theorems relied very much on the very special structure theory of symmetric spaces. Therefore, it may be surprising that such an isomorphism can be constructed for every algebraic variety X carrying an action of a connected reductive group G. Because invariants and centers behave well under field extensions, we assume from now on, that the base field k is algebraically closed of characteristic zero. Let me first explain the case, where X is a smooth, affine G-variety. Here, I obtained the most complete results. This covers all linear actions on a vector space as well as all homogeneous spaces G/H where H is reductive, in particular symmetric varieties. Consider the algebra of (algebraic) linear differential operators D(X). We are interested in the subalgebra D(X) of invariant operators and in particular, in its center Z(X).