Invariant differential operators and an homomorphism of Harish-Chandra

Abstract

Let g be a reductive complex Lie algebra, with adjoint group G, Cartan subalgebra h and Weyl group W . Then G acts naturally on the algebra of polynomial functions O(g) and hence on the ring of differential operators with polynomial coefficients, D(g). Similarly, W acts on h and hence on D(h). In [HC2], Harish-Chandra defined an algebra homomorphism δ : D(g) → D(h) . Recently, Wallach proved that, if g has no factors of type E6, E7 or E8, then this map δ is surjective [Wa, Theorem 3.1]. The significance of Wallach’s result is that it enables him to give an easy proof of an important theorem of Harish-Chandra about invariant distributions and to give an elegant new approach to the Springer correspondence. The main aim of this paper is to give an elementary proof of [Wa, Theorem 3.1] that also works for all reductive Lie algebras. Set