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

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

2012-01-09

arxiv.org/abs/1201.1874v1

Mathematics

Number Theory

68 pages

Scientific paper

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.

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