Harish-Chandra series of modules and modular Howe correspondence
type: Item
instance of: scholarly article
title: Harish-Chandra series of modules and modular Howe correspondence
zbMATH Open document ID: 0851.20047
published in: Journal of Algebra
publication date: 1994-07-07T00:00:00Z
review text: Let \(G\) be a reductive algebraic group defined over the finite field \(\mathbb{F}_q\) and let \(K\) be a local field of characteristic zero with residue field \(k\) of characteristic \(\ell\), where \((\ell,q)=1\) and \(q\) is odd. The author begins by defining induction and restriction functors \(R^G_L\), \(^*R^G_L\) for any 1-split Levi component \(L\) (a Levi component \(L\) is 1-split if it is a rational Levi subgroup of a rational parabolic subgroup \(P\) of \(G\)). Then she defines ``Harish-Chandra series'' of simple \(kG\)-modules; this definition is consistent with the work of \textit{G. Hiss} [J. Lond. Math. Soc., II. Ser. 48, No. 2, 219-228 (1993; Zbl 0814.20004)]. The Harish-Chandra series are parametrized by simple cuspidal \(kL\)-modules. Related work has been done by \textit{R. Dipper} and \textit{J. Du} [J. Reine Angew. Math. 437, 101-130 (1993; Zbl 0768.20004)]. The second part of the paper gives an analogue for the functors \(R^G_L\), \(^*R^G_L\) of the results of \textit{D. Benson} and \textit{R. A. Parker} [J. Algebra 87, 290-331 (1984; Zbl 0539.20009)]. It shows that the Green ring of \(G\) is a direct sum of \(\text{image}(R^G_L)\) and \(\text{ker}(^*R^G_L)\). It requires a new method of proof, since \(R^G_L\) and \(^*R^G_L\) do not satisfy Frobenius reciprocity. If \(G\), \(G'\) are an irreducible reductive dual pair defined over \(\mathbb{F}_q\), it is possible to define the Weil representation. The third part of the article uses it to establish a ``modular Howe correspondence'' from \(kG\)-modules to \(kG'\)-modules. Under certain conditions, this correspondence is shown to preserve cuspidal modules.
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C11
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 11S37
zbMATH DE Number: 589961
zbMATH Keywords: modular Howe correspondence
zbMATH Keywords: reductive algebraic groups
zbMATH Keywords: induction and restriction functors
zbMATH Keywords: Levi components
zbMATH Keywords: rational Levi subgroups
zbMATH Keywords: rational parabolic subgroups
zbMATH Keywords: Harish-Chandra series
zbMATH Keywords: simple cuspidal modules
zbMATH Keywords: Green rings
zbMATH Keywords: Frobenius reciprocity
zbMATH Keywords: irreducible reductive dual pairs
zbMATH Keywords: Weil representations
zbMATH Keywords: cuspidal modules
author: Anne-Marie Aubert
reviewed by: Joe Repka
MaRDI profile type: Publication
full work available at URL: jabr.1994.1132
OpenAlex ID: W2017776044
DOI: 10.1006/JABR.1994.1132
description: scientific article; zbMATH DE number 589961