Eigendistributions for orthogonal groups and representations of symplectic groups

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

Eigendistributions for orthogonal groups and representations of symplectic groups Academic Article