A criterion for integrability of matrix coefficients with respect to a symmetric space

TL;DR

This paper establishes a criterion for the integrability of matrix coefficients over symmetric spaces in p-adic groups, linking it to exponents and generalizing Casselman's criterion, with implications for temperedness and discrete series representations.

Contribution

It introduces a new criterion for G^θ-integrability of matrix coefficients based on exponents, extending classical results and applying to symmetric spaces and discrete series.

Findings

Certain symmetric spaces are strongly tempered.

Matrix coefficients of all discrete series are G^θ-integrable.

The criterion generalizes Casselman's square-integrability condition.

Abstract

Let $G$ be a reductive group and $\theta$ an involution on $G$, both defined over a $p$-adic field. We provide a criterion for $G^\theta$-integrability of matrix coefficients of representations of $G$ in terms of their exponents along $\theta$-stable parabolic subgroups. The group case reduces to Casselman's square-integrability criterion. As a consequence we assert that certain families of symmetric spaces are strongly tempered in the sense of Sakellaridis and Venkatesh. For some other families our result implies that matrix coefficients of all irreducible, discrete series representations are $G^\theta$-integrable.