Distinction of the Steinberg representation with respect to a symmetric pair

Let G be a reductive group over a non-archimedean local field F of odd residual characteristic, 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 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, 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-Courtès on Prasad's conjecture. The basic idea is to realize St 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 Bruhat-Tits buildings. This is a joint work with Chuijia Wang.