A note on local periods for supercuspidal representations

TL;DR

This paper demonstrates that, under mild conditions, local periods for supercuspidal representations of p-adic groups can be explicitly constructed via integration of matrix coefficients over a spherical subgroup.

Contribution

It provides a new method to construct local periods for supercuspidal representations using matrix coefficient integration, under mild assumptions.

Findings

Local periods can be constructed by integrating matrix coefficients.

The method applies to supercuspidal representations of p-adic groups.

Under mild assumptions, the construction is valid.

Abstract

Let $G$ be a $p$-adic reductive group and $H$ a unimodular spherical subgroup of $G$. Let $\pi$ be a unitary supercuspidal representation of $G$. In this note, under a mild assumption, we show that local periods in $Hom_H(\pi,\mathbb{C})$ can be constructed by integrating the matrix coefficients of $\pi$ over $H$.