Invariant differential operators on symplectic grassmann manifolds

LetM _2n,r denote the vector space of real or complex2n×r matrices with the natural action of the symplectic group Sp _2n , and letG=G _n,r =Sp _2n ×M _2n,r denote the corresponding semi-direct product. For any integerp with 0≤p≤n−1, letH denote the subgroupG _p,r ×Sp _2n−2p ofG. We explicitly compute the algebra of left invariant differential operators onG/H, and we show that it is a free algebra if and only ifr≤2n−2p+1. We also give orthogonal analogues of these results, generalizing those of Gonzalez and Helgason [3]. Partially supported by NSF grant DMS-9101358 This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990.