Harmonics on Stiefel Manifolds and Generalized Hankel Transforms

In this note we announce an extension of these theorems to the setting of Stiefel manifolds and matrix space. Our work makes it possible to construct holomorphic discrete series representations for the real symplectic group by decomposing a tensor product of certain projective representations introduced earlier by Shale and Weil. (See Weil [11] and also Shalika [10].) Proofs of the results announced here and their application to the construction of discrete series will appear elsewhere. We let Mnm denote then x m real matrix space, S w,m the Stiefel manifold of matrices VeMnm such that VV = Im9 and Pm the cone of m x m positive-definite symmetric matrices. The rotation group SO(n) acts on S"' and Mnttn by left matrix multiplication so that S"' m s SO(n)/SO(n m). Corresponding to the decomposition Mnm = S w,m x Pm we have the integral formula