Finite multiplicities beyond spherical spaces

TL;DR

This paper generalizes the concept of sphericity in the context of real reductive groups, establishing conditions under which the multiplicities of Schwartz functions on homogeneous spaces remain finite for certain representations.

Contribution

It introduces a new generalized sphericity condition that ensures finite multiplicities beyond the classical spherical case for small irreducible representations.

Findings

Finite multiplicities are guaranteed under the new sphericity generalization.

The results extend the class of homogeneous spaces with finite multiplicities.

The approach applies to small irreducible representations of real reductive groups.

Abstract

Let $G$ be a real reductive algebraic group, and let $H\subset G$ be an algebraic subgroup. It is known that the action of $G$ on the space of functions on $G/H$ is "tame" if this space is spherical. In particular, the multiplicities of the space $\mathcal{S}(G/H)$ of Schwartz functions on $G/H$ are finite in this case. In this paper we formulate and analyze a generalization of sphericity that implies finite multiplicities in $\mathcal{S}(G/H)$ for small enough irreducible representations of $G$.