(GL(2n,C),SP(2n,C)) is a Gelfand Pair
Abstract
We prove that (GL2n(C),Sp2n(C)) is a Gelfand pair. More precisely, we show that for an irreducible smooth admissible Frechet representation (π,E) of GL2n(C) the space of continuous functionals HomSp2n()(E,C) is at most one dimensional. For this we show that any distribution on GL2n(C) invariant with respect to the double action Sp2n(C) × Sp2n(C) is transposition invariant. Such a result was previously proven for p-adic fields by M. Heumos and S. Rallis.
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