Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

• Avraham Aizenbud ^[2] ; Dmitry Gourevitch ^[2] ; Andrei Minchenko ^[1]
  1. [1] University of Vienna

     University of Vienna

     Innere Stadt, Austria

     
  2. [2] Faculty of Mathematics and Computer Science
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 22, Nº. 4 (Special Issue: The Mathematics of Joseph Bernstein), 2016, págs. 2325-2345
• Idioma: inglés
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• Resumen

  • We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein–Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group GG defined over RR , the space of G(R)G(R) -equivariant distributions on the manifold of real points of any algebraic GG -manifold XX is finite-dimensional if GG has finitely many orbits on XX .