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arXiv · 0810.1853

Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R))

Avraham Aizenbud·Eitan Sayag
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Abstract

We study invariant distributions on the tangent space to a symmetric space. We prove that an invariant distribution with the property that both its support and the support of its Fourier transform are contained in the set of non-distinguished nilpotent orbits, must vanish. We deduce, using recent developments in the theory of invariant distributions on symmetric spaces that the symmetric pair (GL(2n,R),Sp(2n,R)) is a Gelfand pair. More precisely, we show that for any irreducible smooth admissible Frechet representation $(π,E)$ of GL(2n,R) the space of continuous functionals $Hom_{Sp_{2n}(R)}(E,C)$ is at most one dimensional. Such a result was previously proven for p-adic fields in [HR] for the field of complex numbers in [S].

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Avraham Aizenbud, Eitan Sayag. 2008-10-10. Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R)). https://arxiv.org/abs/0810.1853

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