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

Abstract

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 G defined over R, the space of G(R)-equivariant distributions on the manifold of real points of any algebraic G-manifold X is finite-dimensional if G has finitely many orbits on X . Mathematics Subject Classification

Erratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

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The following changes to the main results of [1] are necessary:(1) In Theorem A and Corollary B the following assumption is required: the number of orbits of the complexification H C on G C /P C is finite, where P is a minimal parabolic subgroup of G. (2) In Theorem C the following additional assumption is required: the number of orbits of H C on X C is finite.Presently we do not know whether these results hold without the additional assumptions.The source of the mistake is in [2], where the expressions "real algebraic groups" and "real algebraic manifolds" are ambiguous. Moreover, a mistake in [2, Definition 1.

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“…The source of the mistake is in [2], where the expressions "real algebraic groups" and "real algebraic manifolds" are ambiguous. Moreover, a mistake in [2, Definition 1.…”

“…(1) In Theorem A and Corollary B the following assumption is required: the number of orbits of the complexification H C on G C /P C is finite, where P is a minimal parabolic subgroup of G. (2) In Theorem C the following additional assumption is required: the number of orbits of H C on X C is finite.…”

“…Moreover, a mistake in [2, Definition 1. In [2] the terms "real algebraic groups" and "real algebraic manifolds" sometimes mean algebraic groups and manifolds defined over R, and sometimes real points of such. Those two meanings are not equivalent; in particular, the statement that an algebraic group G defined over R acts on an algebraic manifold X with finitely many orbits implies the statement that G(R) acts on X (R) with finitely many orbits, but is not equivalent to it.…”

“…Rather, it is equivalent to the stronger statement that G(C) has finitely many orbits on X (C). In particular, in [2, Theorem D] one needs the stronger assumption (only then it follows from the BernsteinKashiwara theorem, [2, Thm 3.2.2]), see [3]. We do not know whether this theorem holds under the weaker assumption.…”