Local periods for discrete series representations

TL;DR

This paper investigates how to explicitly construct local periods for discrete series representations of p-adic groups in symmetric pairs by integrating matrix coefficients over the subgroup, providing new insights into harmonic analysis on p-adic symmetric spaces.

Contribution

It demonstrates that for certain symmetric pairs, local periods can be explicitly constructed through matrix coefficient integration, advancing understanding of representation theory in p-adic contexts.

Findings

Local periods can be constructed via matrix coefficient integration.

Applicable to specific types of symmetric pairs.

Provides explicit methods for harmonic analysis on p-adic symmetric spaces.

Abstract

Let $(G,H)$ be a symmetric pair over a $p$-adic field and $\pi$ a discrete series representation of $G$. In this paper, for some type of symmetric pairs $(G,H)$, we show that local periods in $Hom_H(\pi,\mathbb{C})$ can be constructed by integrating the matrix coefficients of $\pi$ over $H$.