Invariant differential operators on symplectic grassmann manifolds

Abstract

Let M 2n,r denote the vector space of real or complex 2n×r matrices with the natural action of the symplectic group Sp 2n, and let G=G n,r =Sp 2n ×M 2n,r denote the corresponding semi-direct product. For any integer p with 0≤p≤n-1, let H denote the subgroup G p,r ×Sp 2n-2p of G. We explicitly compute the algebra of left invariant differential operators on G/H, and we show that it is a free algebra if and only if r≤2n-2p+1. We also give orthogonal analogues of these results, generalizing those of Gonzalez and Helgason [3]. © 1994 Springer-Verlag.