Distinction of the Steinberg representation with respect to a symmetric pair

TL;DR

This paper investigates when the Steinberg representation of a reductive group over a non-archimedean local field is distinguished by a symmetric subgroup, providing bounds, explicit bases, and a classification in specific cases.

Contribution

It introduces a method to analyze the distinction of the Steinberg representation via harmonic cochains, establishes bounds on the dimension of distinguished spaces, and classifies cases for general linear and orthogonal groups.

Findings

Upper bound on the dimension of the distinguished space.

Explicit basis construction using Poincaré series.

Complete classification for GL and orthogonal subgroup cases.

Abstract

Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $\theta$ be an involution of $G$ over $K$, and let $H$ be the connected component of $\theta$-fixed subgroup of $G$ over $K$. By realizing the Steinberg representation of $G$ as the $G$-space of complex smooth harmonic cochains following the idea of Broussous--Court\`es, we study its space of distinction by $H$ as a finite dimensional complex vector space. We give an upper bound of the dimension, and under certain conditions, we show that the upper bound is sharp by explicitly constructing a basis using the technique of Poincar\'e series. Finally, we apply our general theory to the case where $G$ is a general linear group and $H$ a special orthogonal subgroup, which leads to a complete classification result.