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

TL;DR

This paper proves that relative characters of admissible representations are holonomic distributions, leading to analytic properties and applications in bounding multiplicities for spherical pairs, with implications in both real and p-adic cases.

Contribution

It establishes the holonomicity of relative characters for spherical pairs and develops new algebraic geometric methods to analyze multiplicity bounds.

Findings

Relative characters are holonomic distributions.

Restriction of relative characters is analytic on dense subsets.

Finite-dimensionality of distribution spaces under certain group actions.

Abstract

In this paper, 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 from [KO13,KS16] on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. In order to deduce this application we prove 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 $\mathbb{R}$, the space of $G(\mathbb{R})$-equivariant distributions on the manifold of real points of any algebraic $G$-manifold $X$ is finite-dimensional if $G$ has finitely many orbits on $X$.