Invariant distributions on non-distinguished nilpotent orbits with application to the gelfand property of (GL 2n(ℝ), Sp 2n(ℝ))

Avraham Aizenbud, Eitan Sayag

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3 Scopus citations

Abstract

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 Fréchet representation (π,E) of (GL 2n(ℝ) the space of continuous functionals Hom Sp2n(ℝ)(E,ℂ) is at most one dimensional. Such a result was previously proven for p-adic fields in M. J. Heumos and S. Rallis, Symplectic-Whittaker models for Gl n , Pacific J. Math. 146 (1990), 247-279, and for ℂ in E. Sayag, (GL 2n(ℝ), Sp 2n(ℝ)) is a Gelfand pair, arXiv:0805.2625 [math.RT].

Original languageEnglish
Pages (from-to)137-153
Number of pages17
JournalJournal of Lie Theory
Volume22
Issue number1
StatePublished - 6 Feb 2012

Keywords

  • Co-isotropic sub-variety
  • Gelfand pair
  • Invariant distribution
  • Multiplicity one
  • Non-distinguished orbits
  • Symmetric pair
  • Symplectic group

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