Let $G$ be a reductive group over a non-archimedean local field $F$ of residual characteristic $p\neq 2$, let $\theta$ be an involution of $G$ over $F$ and let $H$ be the connected component of the $\theta$-fixed subgroup of $G$. We are interested in the problem of distinction of the Steinberg representation $St_{G}$ of $G$ restricted to $H$. More precisely, first we give a reasonable upper bound of the dimension of the complex vector space $$Hom_H(St_G|_H,C)$$ which was previously known to be finite, and secondly we calculate this dimension for special symmetric pairs $(G,H)$. For instance, the most interesting case for us is when $G$ is a general linear group and $H$ is an orthogonal subgroup of $G$. Our method follows from the previous results of Broussous--Court\`es on Prasad's conjecture. The basic idea is to realize $St_{G}$ as the $G$-space of complex harmonic cochains on the Bruhat--Tits building of $G$. Thus the problem is somehow reduced to the combinatorial geometry of the Bruhat--Tits building. This is a joint work with Chuijia Wang.

# Cable Car Algebra Seminar: Distinction of the Steinberg representation with respect to a symmetric pair

Abstract: