The Bernstein center in natural characteristic

TL;DR

This paper investigates the structure of the Bernstein center for certain locally profinite groups over fields of positive characteristic, establishing isomorphisms and explicit descriptions under specific conditions related to the group's structure.

Contribution

It proves an isomorphism between the Bernstein centers of a group and its center for groups with particular properties, and explicitly describes this center as a completion of the group ring.

Findings

Isomorphism of Bernstein centers under specified conditions

Explicit description of the Bernstein center as a completed group ring

Results apply to groups of points of connected algebraic groups over local fields

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 $\mathfrak{Z}(G)$ 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 $\mathfrak{Z}(G) = k[Z(G)]$.