Universal modules in natural characteristic for finite reductive groups.
type: Item
instance of: scholarly article
title: Universal modules in natural characteristic for finite reductive groups.
zbMATH Open document ID: 1348.20051
author: Rachel Ollivier
author: Vincent Sécherre
publication date: 2015-10-20T00:00:00Z
review text: Let \(k\) be a finite field with characteristic denoted by \(p\), let \(\mathbf G\) be a connected reductive algebraic group defined over \(k\), and let \(\mathbf U\) be the unipotent radical of a Borel subgroup \(\mathbf B\) defined over \(k\). Let \(G\), \(U\) and \(B\) be the \(k\)-rational points of \(\mathbf G\), \(\mathbf U\) and \(\mathbf B\), respectively. Define \(C\) as the space of functions on \(G\) with values in an algebraic closure \(\overline k\) of \(k\) that are invariant by left translations by \(U\). Then \(C\) is called the \textit{universal module} for \(G\). The group \(G\) acts by right translation on \(C\). The commuting algebra \(H\) of the resulting representation of \(G\) is the Hecke algebra of \(G\) with respect to \(U\). The aim of the article is the study of the representations of \(G\) with coefficients in \(\overline k\), and the links of the latters with \(H\)-modules. It is motivated by (and used in) the study of mod \(p\) representations of \(p\)-adic reductive groups. Since \(C\) is both a representation of \(G\) and a left \(H\)-module, there is a functor \(\mathbf F\colon V\mapsto\Hom_G(C,V)\) from the category of \(\overline k\)-representations of \(G\) to the category of right \(H\)-modules. In general this functor is not exact, because the order of \(U\) is not invertible in \(\overline k\). Nevertheless, it induces a bijection between the isomorphism classes of irreducible representations of \(G\) and those of simple \(H\)-modules. It admits as right adjoint the functor \(E\mapsto E\otimes_C H\). The first main result proved in the article is the equivalence of the flatness of the \(H\)-module \(C\) with the fact that the functor \(\mathbf F\) induces an equivalence between the category of finite dimensional \(\overline k\)-representations of \(G\) that are generated by their \(U\)-invariant vectors and the category of finite dimensional right \(H\)-modules. This result is part of Proposition 2.5, which provides also sever…
reviewed by: Anne-Marie Aubert
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 6496545
zbMATH Keywords: modular representations
zbMATH Keywords: finite reductive groups
zbMATH Keywords: Hecke algebras
zbMATH Keywords: categories of representations
zbMATH Keywords: categories of modules
MaRDI profile type: Publication
full work available at URL: aif.2936
OpenAlex ID: W2561138292
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DOI: 10.5802/AIF.2936
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 6496545