Invariant differential operators on the tangent space of some symmetric spaces

Let g be a complex, semisimple Lie algebra, with an involutive automorphism upsilon and set t = Ker(upsilon - I), p = Ker(upsilon + I) We consider the differential operators, D(p)(K), on p that are invariant under the action of the adjoint group K of t. Write tau : it --> Der O(p) for the differential of this action. Then we prove, for the class of symmetric pairs (g,t) considered by Sekiguchi, that {d is an element of D(p) : d(O(p)(K)) = 0} = D(p)tau(t). An immediate consequence of this equality is the following result of Sekiguchi : Let (g(0), t(0)) be a real form of one of these symmetric pairs (g,t), and suppose that T is a K-0-invariant eigendistribution on that is supported on the singular set. Then, T = O. In the diagonal case (g,t) = (g' + g', g') this is a well-known result due to Harish-Chandra.