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

TL;DR

This paper analyzes the structure of the Bernstein center in the category of smooth modules over the group algebra of GL_n over a p-adic field, revealing its block decomposition, algebraic properties, and connection to supercuspidal supports.

Contribution

It provides a detailed description of the Bernstein center for smooth modules over W(k)[GL_n(F)], including block decomposition, algebraic structure, and explicit subalgebras, extending classical characteristic zero results.

Findings

Each block's center is a reduced, finite type, l-torsion free W(k)-algebra.

The k-points of the center correspond to supercuspidal supports.

An explicit subalgebra of the center acts on simple objects, mirroring classical Bernstein center behavior.

Abstract

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 block is a reduced, finite type, l-torsion free W(k)-algebra. Moreover, the k-points of the center of each 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 subalgebra on the simple objects of the block, in terms of the description of the classical "characteristic zero" Bernstein center.