The Bernstein center in natural characteristic
Let $G$ be a locally profinite group and let $k$ be a field of positive characteristic $p$. Let $Z(G)$ denote the center of $G$ and let $\mathfrak{Z}(G)$ denote the Bernstein center of $G$, that is, the $k$-algebra of natural endomorphisms of the identity functor on the category of smooth $k$-linear...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Springer
2023
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| _version_ | 1826313209386106880 |
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| author | Ardakov, K Schneider, P |
| author_facet | Ardakov, K Schneider, P |
| author_sort | Ardakov, K |
| collection | OXFORD |
| description | Let $G$ be a locally profinite group and let $k$ be a field of positive
characteristic $p$. Let $Z(G)$ denote the center of $G$ and let
$\mathfrak{Z}(G)$ denote the Bernstein center of $G$, that is, the $k$-algebra
of natural endomorphisms of the identity functor on the category of smooth
$k$-linear representations of $G$. We show that if $G$ contains an open pro-$p$
subgroup but no proper open centralisers, then there is a natural isomorphism
of $k$-algebras $\mathfrak{Z}(Z(G)) \xrightarrow{\cong} \mathfrak{Z}(G)$. We
also describe $\mathfrak{Z}(Z(G))$ explicitly as a particular completion of the
abstract group ring $k[Z(G)]$. Both conditions on $G$ are satisfied whenever
$G$ is the group of points of any connected smooth algebraic group defined over
a local field of residue characteristic $p$. In particular, when the algebraic
group is semisimple, we show that $\mathfrak{Z}(G) = k[Z(G)]$. |
| first_indexed | 2024-03-07T07:46:00Z |
| format | Journal article |
| id | oxford-uuid:ce0edee9-9e12-4cc1-94d3-22541ee3547f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:07:54Z |
| publishDate | 2023 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:ce0edee9-9e12-4cc1-94d3-22541ee3547f2024-06-06T09:25:53ZThe Bernstein center in natural characteristicJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ce0edee9-9e12-4cc1-94d3-22541ee3547fEnglishSymplectic ElementsSpringer2023Ardakov, KSchneider, PLet $G$ be a locally profinite group and let $k$ be a field of positive characteristic $p$. Let $Z(G)$ denote the center of $G$ and let $\mathfrak{Z}(G)$ denote the Bernstein center of $G$, that is, the $k$-algebra of natural endomorphisms of the identity functor on the category of smooth $k$-linear representations of $G$. We show that if $G$ contains an open pro-$p$ subgroup but no proper open centralisers, then there is a natural isomorphism of $k$-algebras $\mathfrak{Z}(Z(G)) \xrightarrow{\cong} \mathfrak{Z}(G)$. We also describe $\mathfrak{Z}(Z(G))$ explicitly as a particular completion of the abstract group ring $k[Z(G)]$. Both conditions on $G$ are satisfied whenever $G$ is the group of points of any connected smooth algebraic group defined over a local field of residue characteristic $p$. In particular, when the algebraic group is semisimple, we show that $\mathfrak{Z}(G) = k[Z(G)]$. |
| spellingShingle | Ardakov, K Schneider, P The Bernstein center in natural characteristic |
| title | The Bernstein center in natural characteristic |
| title_full | The Bernstein center in natural characteristic |
| title_fullStr | The Bernstein center in natural characteristic |
| title_full_unstemmed | The Bernstein center in natural characteristic |
| title_short | The Bernstein center in natural characteristic |
| title_sort | bernstein center in natural characteristic |
| work_keys_str_mv | AT ardakovk thebernsteincenterinnaturalcharacteristic AT schneiderp thebernsteincenterinnaturalcharacteristic AT ardakovk bernsteincenterinnaturalcharacteristic AT schneiderp bernsteincenterinnaturalcharacteristic |