Eigendistributions for orthogonal groups and representations of symplectic groups
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We consider the action of H = O(p, q) on the matrix space M p+q,n (ℝ). We study a certain orbit script O sign of H in the null cone N ⊆ M p+q,n (ℝ) 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 G̃ = Sp(2n, ℝ)̃ -cyclic module generated by dv script 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.
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