On mod p local-global compatibility for GL_n(Q_p) in the ordinary case

Abstract

Let p be a prime number, n>2 an integer, and F a CM field in which p splits completely. Assume that a continuous automorphic Galois representation r¯¯:Gal(Q¯¯¯¯/F)→GLn(F¯¯¯¯p) is upper-triangular and satisfies certain genericity conditions at a place w above~p, and that every subquotient of r¯¯|Gal(Q¯¯¯¯¯p/Fw) of dimension >2 is Fontaine-Laffaille generic. In this paper, we show that the isomorphism class of r¯¯|Gal(Q¯¯¯¯¯p/Fw) is determined by GLn(Fw)-action on a space of mod p algebraic automorphic forms cut out by the maximal ideal of a Hecke algebra associated to r¯¯. In particular, we show that the wildly ramified part of r¯¯|Gal(Q¯¯¯¯¯p/Fw) is determined by the action of Jacobi sum operators (seen as elements of Fp[GLn(Fp)]) on this space.