Modular representations of PGL2 and automorphic forms for shimura curves

Abstract

In this paper, we study certain infinite dimensional representations of the general linear group GL2 over a local field K, in vector spaces over K. These representations are closely related to the theory of analytic functions on the p-adic upper half plane over K. They were originally introduced by Morita [Mor] , who studied them intensively in the case when the characteristic of K is zero. Morita 's work has subsequently been greatly extended by Schneider and Stuhler [SS] and Schneider [S] to representations of GL,(K) and higher p-adic symmetric spaces. In this paper, we re-consider some of Morita 's representations of GL2 from the point of view of integral and modular representation theory. It seems quite natural to study p-adic representations by reducing them modp, and we supply a technique for doing this. In particular, suppose that Ydenotes the Bruhat Tits tree of PGLz(K). We consider the representations of PGL2(K) on the harmonic functions on W of e v e n weight that is, the harmonic functions on the edges of J taking values in even symmetric powers of the standard representation of GL2. In this paper, we find an invariant integral lattice in these representations, and we describe the modular representations of PGL2(K) obtained by reducing this lattice modp. The modular representations obtained from Morita 's representations are quite simple and natural objects, although they seem a bit peculiar at first. Suppose that F is the residue field of K, that V is a vector space over F, and that p :GL2(F) ~ End(V) is a finite-dimensional (modular) representation. Notice that p may be viewed in a natural way as a modular representation of GL/(R), where R is the valuation ring of K. We show that a typical infinite dimensional representation of Morita's, when reduced mod p, has a filtration whose factors are isomorphic to "I"~CL2~R~WJ.T~n~L2~)t~ Note that there are only finitely many irreducible, finite dimensional modular representations of GLz(F); the appearance of these finitely many