Invariant differential operators on a reductive Lie algebra and Weyl group representations

Abstract

Let V be a finite-dimensional vector space, and let G be a subgroup of GL(V). Set D(V) equal to the algebra of differential operators on V with polynomial coefficients and D(V)G equal to the G invariants in D(V) . If g is a reductive Lie algebra over C then t c g is a Cartan subgroup of g, and if G is the adjoint group of g then W is the Weyl group of (g, [), Harish-Chandra introduced an algebra homomorphism, (5, of D(g) G to D(t)w [H3]. a is given by the obvious restriction mapping on the subalgebra of invariant polynomials and on the invariant constant coefficient differential operators, and ker( is