Eigendistributions for orthogonal groups and representations of symplectic groups

Abstract

We consider the action of H = O(p, q) on the matrix space Mp+q,n(ℝ). We study a certain orbit script O sign of H in the null cone N ⊆ Mp+q,n(ℝ) which supports an eigen-distribution dvscript O sign for H. Using some identities of Capelli type developed in the Appendix, we determine the structure of G̃ = Sp(2n, ℝ)̃ -cyclic module generated by dvscript O sign under the oscillator representation of double-struck G sigñ (the metaplectic cover of double-struck G sign = Sp(2n(p + q), ℝ)). Applications to local theta correspondence and a generalized Huygens' Principle are given.