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We consider the category of smooth $W(k)[GL_n(F)]$-modules, where F is ap-adic field and k is an algebraically closed field of characteristic ldifferent from p. We describe a factorization of this category into blocks, andshow that the center of each block is a reduced, finite type, l-torsion freeW(k)-algebra. Moreover, the k-points of the center of each block are inbijection 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 theaction of this subalgebra on the simple objects of the block, in terms of thedescription of the classical "characteristic zero" Bernstein center.

The Bernstein center of the category of smooth $W(k)[GL_n(F)]$-modules