Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 22 (2012), No. 1, 137--153Copyright 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. aizenr@gmail.com Eitan Sayag Dept. of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel sayage@math.bgu.ac.il [Abstract-pdf] \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 [ Fulltext-pdf  (410  KB)] for subscribers only.