表示论与数论讨论班—Distinction of the Steinberg representation with respect to a symmetric pair
摘要：Let G be a reductive group over a non-archimedean local field F of residual characteristic p different from 2, let θ be an involution of G over F and let H be the connected component of the θ-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, 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–Courtes on Prasad’s conjecture. The basic idea is to realize StG 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 building. This is a joint work with Chuijia Wang.