Distinction of the Steinberg representation with respect to split symmetric subgroups

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 do that both over a $p$ adic field and over a finite field. We relate these distinction problems to problems about determining the existence of a non zero harmonic function on certain hyper graphs related to $X = G/H$. Under these assumptions, 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 Langalnds parameter factors through the dual group of $X$.