Geodesic
Invariant Distributions on Non-Distinguished Nilpotent Orbits with Application to the Gelfand Property of (GL2n(R),Sp2n(R))
Avraham Aizenbud1 ; Eitan Sayag2
1Dept. of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.
2Dept. of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Journal of Lie Theory, Tome 22 (2012) no. 1, pp. 137-153

Voir la notice de l'article provenant de la source Heldermann Verlag

\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

Avraham Aizenbud  1   ; Eitan Sayag  2

1 Dept. of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.
2 Dept. of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel 
Avraham Aizenbud; Eitan 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/JOLT_2012_22_1_a4/
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     author = {Avraham Aizenbud and Eitan 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},
     year = {2012},
     volume = {22},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2012_22_1_a4/}
}
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