(GL(2n,C),SP(2n,C)) is a Gelfand Pair

Details (GL(2n,C),SP(2n,C)) is a Gelfand Pair (GL(2n,C),SP(2n,C)) is a Gelfand Pair

2008-05-19

arxiv.org/abs/0805.2625v1

Mathematics

Representation Theory

10 pages

Scientific paper

We prove that (GL_{2n}(C),Sp_{2n}(C)) is a Gelfand pair. More precisely, we show that for an irreducible smooth admissible Frechet representation (\pi,E) of GL_{2n}(C) the space of continuous functionals Hom_{Sp_{2n}(\cc)}(E,C) is at most one dimensional. For this we show that any distribution on GL_{2n}(C) invariant with respect to the double action Sp_{2n}(C) \times Sp_{2n}(C) is transposition invariant. Such a result was previously proven for p-adic fields by M. Heumos and S. Rallis.

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