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 |
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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 |
| _version_ | 1827994914506407936 |
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| author | DAVID HELM |
| author_facet | DAVID HELM |
| author_sort | DAVID HELM |
| collection | DOAJ |
| description | 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]. |
| first_indexed | 2024-04-10T04:47:19Z |
| format | Article |
| id | doaj.art-9415569f51174e628d23132741650d2e |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:19Z |
| publishDate | 2016-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-9415569f51174e628d23132741650d2e2023-03-09T12:34:41ZengCambridge University PressForum of Mathematics, Sigma2050-50942016-01-01410.1017/fms.2016.10THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH $W(k)[\text{GL}_{n}(F)]$ -MODULESDAVID HELM0Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK;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].https://www.cambridge.org/core/product/identifier/S2050509416000104/type/journal_article11F3311F70 (primary)22E50 (secondary) |
| spellingShingle | DAVID HELM THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH $W(k)[\text{GL}_{n}(F)]$ -MODULES Forum of Mathematics, Sigma 11F33 11F70 (primary) 22E50 (secondary) |
| title | THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH $W(k)[\text{GL}_{n}(F)]$ -MODULES |
| title_full | THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH $W(k)[\text{GL}_{n}(F)]$ -MODULES |
| title_fullStr | THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH $W(k)[\text{GL}_{n}(F)]$ -MODULES |
| title_full_unstemmed | THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH $W(k)[\text{GL}_{n}(F)]$ -MODULES |
| title_short | THE BERNSTEIN CENTER OF THE CATEGORY OF SMOOTH $W(k)[\text{GL}_{n}(F)]$ -MODULES |
| title_sort | bernstein center of the category of smooth w k text gl n f modules |
| topic | 11F33 11F70 (primary) 22E50 (secondary) |
| url | https://www.cambridge.org/core/product/identifier/S2050509416000104/type/journal_article |
| work_keys_str_mv | AT davidhelm thebernsteincenterofthecategoryofsmoothwktextglnfmodules AT davidhelm bernsteincenterofthecategoryofsmoothwktextglnfmodules |