On the generalized Steinberg representation
type: Item
instance of: scholarly article
title: On the generalized Steinberg representation
zbMATH Open document ID: 1018.20038
author: Nilabh Sanat
published in: Journal of Combinatorial Theory. Series A
publication date: 2002-07-04T00:00:00Z
review text: Let \(G\) be a connected split reductive group defined over a finite field \(\mathbb{F}_q\). The aim of this paper is to study certain generalizations of the Steiberg representation of \(G(\mathbb{F}_q)\). The author shows that the generalized Steinberg representation \(\text{St}^{P_J}_G\) decomposes as a multiplicity free representation when \(G\) is of type \(A_n\), \(B_n\) and \(C_n\), and \(P_J\) is a maximal parabolic subgroup. He gives an explicit decomposition of \(\text{St}_G^{P_J}\) in these cases. When \(G\) is of type \(D_n\), \(E_6\) and \(E_7\), and \(P_J\) is a maximal parabolic subgroup with Abelian unipotent radical, he proves that \(\text{St}^{P_J}_G\) decomposes as a multiplicity free representation. In the case of \(E_6\) and \(E_7\), he describes the irreducible components of \(\text{St}^{P_J}_G\) using work of \textit{M. Reeder} [Indag. Math., New Ser. 9, No. 3, 431-441 (1998; Zbl 0921.22012)].
reviewed by: Ye Jiachen
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 1763783
zbMATH Keywords: reductive groups
zbMATH Keywords: Steinberg representations
zbMATH Keywords: multiplicity-free representations
zbMATH Keywords: maximal parabolic subgroups
MaRDI profile type: Publication
full work available at URL: jcta.2001.3210
OpenAlex ID: W1965163260
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DOI: 10.1006/JCTA.2001.3210
description: scientific article; zbMATH DE number 1763783