## 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 language | English |
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Pages (from-to) | 137-153 |

Number of pages | 17 |

Journal | Journal of Lie Theory |

Volume | 22 |

Issue number | 1 |

State | Published - 6 Feb 2012 |

## Keywords

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