Invariant Differential Operators on the Tangent Space of Some Symmetric Spaces

Let g be a complex, semisimple Lie algebra, with an involutive automorphism θ and set k = Ker(θ − I), p = Ker(θ + I). We consider the differential operators, D(p) , on p that are invariant under the action of the adjoint group K of k. Write τ : k → Der(p) for the differential of this action. Then we prove, for the class of symmetric pairs (g, k) considered by Sekiguchi [32], that ̆ d ∈ D(p) : d ` O(p)K ́ = 0 ̄ = D(p)τ(k). One significance of this result is that it easily implies the following result of Sekiguchi: Let (g0, k0) be a real form of one of these symmetric pairs (g, k), and suppose that T is a K0-invariant eigendistribution on p0 that is supported on the singular set. Then, T = 0. In the diagonal case (g, k) = (g′⊕ g′, g′) this is a well-known result due to Harish-Chandra.