Distinction of the Steinberg representation and the dual group of a symmetric space

TL;DR

This paper investigates when the Steinberg representation of a split reductive group is distinguished by a symmetric subgroup, linking this to harmonic functions on hyper-graphs and confirming a local Langlands conjecture.

Contribution

It establishes a criterion for the distinction of the Steinberg representation in terms of Langlands parameters and the dual group of the symmetric space.

Findings

Steinberg representation is H-distinguished iff its Langlands parameter factors through the dual group of X.

Relates distinction problem to existence of harmonic functions on hyper-graphs.

Verifies the relative local Langlands conjecture for the Steinberg representation.

Abstract

We study the distinction of the Steinberg representation of a split reductive group $G$ with respect to a split symmetric subgroup $H \subset G$. We relate this distinction problem to a problem about the existence of a non-zero harmonic function on a certain hyper-graph related to $X = G/H$. We verify the relative local Langlands conjecture for the Steinberg representation by showing that over a $p$-adic field the Steinberg representation is $H$-distinguished if and only if its Langlands parameter factors through the dual group of $X$.