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

We consider the category of smooth W (k)[GLn(F )]-modules, where F is a p-adic field and k is an algebraically closed field of characteristic ` different from p. We describe a factorization of this category into blocks, and show that the center of each such block is a reduced, `-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[GLn(F )]modules that lie in the block. Finally, we describe a large explicit subalgebra of the center of each block and 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 [BD].