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Journal of Lie Theory 22 (2012), No. 1, 137--153
Copyright Heldermann Verlag 2012

Invariant Distributions on Non-Distinguished Nilpotent Orbits with Application to the Gelfand Property of (GL2n(R),Sp2n(R))

Avraham Aizenbud
Dept. of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.

Eitan Sayag
Dept. of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel


\def\C{{\mathbb{C}}} \def\R{{\mathbb{R}}} 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 Fr\'echet representation $(\pi,E)$ 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 by M. J. Heumos and S. Rallis [Symplectic-Whittaker models for Gl$_n$, Pacific J. Math. 146 (1990) 247--279], and for $\C$ by the second author [$(GL_{2n}(\C),Sp_{2n}(\C))$ is a Gelfand pair, arXiv:0805.2625, math.RT].

Keywords: Symmetric pair, Gelfand pair, symplectic group, non-distinguished orbits, multiplicity one, invariant distribution, co-isotropic subvariety.

MSC: 20G05, 22E45, 20C99, 46F10

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