
Journal of Lie Theory 22 (2012), No. 1, 137153 Copyright Heldermann Verlag 2012 Invariant Distributions on NonDistinguished Nilpotent Orbits with Application to the Gelfand Property of (GL_{2n}(R),Sp_{2n}(R)) Avraham Aizenbud Dept. of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. aizenr@gmail.com Eitan Sayag Dept. of Mathematics, BenGurion University of the Negev, BeerSheva, Israel sayage@math.bgu.ac.il [Abstractpdf] \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 nondistinguished 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 [SymplecticWhittaker models for Gl$_n$, Pacific J. Math. 146 (1990) 247279], 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, nondistinguished orbits, multiplicity one, invariant distribution, coisotropic subvariety. MSC: 20G05, 22E45, 20C99, 46F10 [ Fulltextpdf (410 KB)] for subscribers only. 