(GL(2n,C),SP(2n,C)) is a Gelfand Pair
Representation Theory
2008-05-20 v1 Number Theory
Abstract
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.
Cite
@article{arxiv.0805.2625,
title = {(GL(2n,C),SP(2n,C)) is a Gelfand Pair},
author = {Eitan Sayag},
journal= {arXiv preprint arXiv:0805.2625},
year = {2008}
}
Comments
10 pages