THE KERNEL OF AN HOMOMORPHISM OF HARISH-CHANDRA

Let g be a reductive, complex Lie algebra, with adjoint group G, let G act on the ring of differential operators D(g) via the adjoint action and write tau : g --> D)(g) for the differential of this action. A cl assic result of Harish-Chandra shows that any invariant differential o perator that kills O(g)(G), the algebra of invariant functions on g, a lso kills all invariant distributions on a real form of g. In this pap er we generalize this result by showing that D(g)tau(g) = {theta is an element of D(g) : theta(O(g)(G)) = 0}. This answers a question raised by Dixmier, by Wallach and by Schwarz.