A distinction criterion for Iwahori-spherical representations

TL;DR

This paper establishes a criterion linking Iwahori-spherical representations of a p-adic group to their distinction properties with respect to a subgroup, via a subgroup of the Iwahori-Hecke algebra.

Contribution

It introduces a new subgroup criterion within the Iwahori-Hecke algebra that characterizes when Iwahori-spherical representations are distinguished by a symmetric subgroup.

Findings

Existence of a subgroup mma of the Iwahori-Hecke algebra al G that detects H-distinguished representations.

An equivalence between H-distinction of Iwahori-spherical representations and mma-distinction of their modules.

Provides a new algebraic criterion for understanding distinction in p-adic representation theory.

Abstract

Let $G/H$ be a Galois symmetric space for an unramified quadratic extension of a locally compact field $F$, where the group $H$ is semisimple, simply connected, defined and split over $F$. We prove that there exists a subgroup $\Gamma = \Gamma (G/H)$ of the group of invertible elements of the Iwahori-Hecke algebra $\mathcal H$ of $G$ such that an Iwahori-spherical representation of $G$ is $H$-distinguished if and only if the corresponding Iwahori-Hecke module is "$\Gamma$-distinguished".