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.
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
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