The Bernstein centre in natural characteristic

DOI10.1134/S0081543823010017zbMATH Open1527.20039arXiv2105.06128MaRDI QIDQ6098036

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Publication date: 9 June 2023

Published in: Proceedings of the Steklov Institute of Mathematics (Search for Journal in Brave)

Abstract: 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 $mathfrakZ(G)$ 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 $mathfrakZ(Z(G))xrightarrowcongmathfrakZ(G)$. We also describe $mathfrakZ(Z(G))$ 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 $mathfrakZ(G)=k[Z(G)]$.

Full work available at URL: https://arxiv.org/abs/2105.06128

Mathematics Subject Classification ID

Representations of Lie and linear algebraic groups over local fields (22E50) Local properties of groups (20E25) Limits, profinite groups (20E18)

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