THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH W(k)[GLn(F)]-MODULES

We consider the category of smooth W(k)[GL(n)(F)]-modules, where F is a p-adic field and k is an algebraically closed field of characteristic l different from p. We describe a factorization of this category into blocks, and show that the center of each such block is a reduced, l-torsion free, finite type W(k)-algebra. Moreover, the k-points of the center of a such a block are in bijection with the possible 'supercuspidal supports' of the smooth k[GL(n)(F)]-modules that lie in the block. Finally, we describe a large explicit subalgebra of the center of each block and give a description of the action of this algebra on the simple objects of the block, in terms of the description of the classical 'characteristic zero' Bernstein center of Bernstein and Deligne [Le 'centre' de Bernstein, in Representations des groups redutifs sur un corps local, Traveaux en cours (ed. P. Deligne) (Hermann, Paris), 1-32].