Distinguished non-Archimedean representations

Abstract

For a symmetric space (G,H), one is interested in understanding the vector space of H-invariant linear forms on a representation π of G. In particular an important question is whether or not the dimension of this space is bounded by one. We cover the known results for the pair (G=R_E/FGL(n),H=GL(n)), and then discuss the corresponding SL(n) case. In this paper, we show that (G=R_E/FSL(n),H=SL(n)) is a Gelfand pair when n is odd. When n is even, the space of H-invariant forms on π can have dimension more than one even when π is supercuspidal. The latter work is joint with Dipendra Prasad.