THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH $W(k)[\text{GL}_{n}(F)]$ -MODULES
We consider the category of smooth $W(k)[\text{GL}_{n}(F)]$ -modules, where $F$ is a...
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2016-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509416000104/type/journal_article |
| Summary: | 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]. |
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| ISSN: | 2050-5094 |