Geodesic
Invariant Distributions on Non-Distinguished Nilpotent Orbits with Application to the Gelfand Property of (GL2n(R),Sp2n(R))
Journal of Lie theory, Tome 22 (2012) no. 1, pp. 137-153.

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\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].
Classification : 20G05, 22E45, 20C99, 46F10
Mots-clés : Symmetric pair, Gelfand pair, symplectic group, non-distinguished orbits, multiplicity one, invariant distribution, co-isotropic subvariety 
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     author = {A. Aizenbud and E. Sayag },
     title = {Invariant {Distributions} on {Non-Distinguished} {Nilpotent} {Orbits} with {Application} to the {Gelfand} {Property} of {(GL\protect\textsubscript{2n}(R),Sp\protect\textsubscript{2n}(R))}},
     journal = {Journal of Lie theory},
     pages = {137--153},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2012},
     url = {http://geodesic.mathdoc.fr/item/JLT_2012_22_1_JLT_2012_22_1_a4/}
}
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A. Aizenbud; E. Sayag . Invariant Distributions on Non-Distinguished Nilpotent Orbits with Application to the Gelfand Property of (GL2n(R),Sp2n(R)). Journal of Lie theory, Tome 22 (2012) no. 1, pp. 137-153. http://geodesic.mathdoc.fr/item/JLT_2012_22_1_JLT_2012_22_1_a4/