THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH -MODULES

Abstract

We consider the category of smooth $W(k)[\text{GL}_{n}(F)]$-modules, where $F$ is a $p$-adic field and $k$ is an algebraically closed field of characteristic $\ell$ different from $p$. We describe a factorization of this category into blocks, and show that the center of each such block is a reduced, $\ell$-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 $k[\text{GL}_{n}(F)]$-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].