Schur algebras of reductive \(p\)-adic groups. I
type: Item
instance of: scholarly article
title: Schur algebras of reductive \(p\)-adic groups. I
zbMATH Open document ID: 1018.22015
DOI: 10.1215/S0012-7094-03-11612-9
author: Marie-France Vigneras
published in: Duke Mathematical Journal
publication date: 2003-09-02T00:00:00Z
review text: This paper concerns representations of the \(p\)-adic group \(G=GL(n,F)\) over an algebraically closed field \(R\) of characteristic not equal to \(p\). It gives a link between the representations of the \(p\)-affine Schur algebra of \(G\) over \(R\) and the smooth \(R\)-representations of \(G\). In the earlier work [\textit{M.-F. Vigneras}, Invent. Math. 144, 177-223 (2001; Zbl 1031.11068)], the author gave the local Langlands \(R\)-correspondence: a parametrization of irreducible \(R\)-representations of \(G\). In this paper, she considers the category of all \(R\)-representations. Let \(\text{B}_{R,\tau}\) be the block parametrized by the parameter \(\tau\). The author considers the unipotent block \(\text{B}_{R,1}\). Let \(I\) be an Iwahori subgroup of \(G\) and let \(\text{ J}_R\) be the annihilator in the Hecke \(R\)-algebra of \(G\) of the \(R\)-representation of \(G\), \(R[I\setminus G]\). The main theorem is the following: There exists an integer \(N\) such that \(\text{ J}^N_R\) annihilates the unipotent block. The category of \(R\)-representations of \(G\) annihilated by \(\text{ J}_R\) is equivalent to the category of modules of the affine Schur algebra of \(G\). The main theorem is interesting because it creates a bridge between the \(p\)-adic groups and the affine quantum groups.
reviewed by: Dubravka Ban
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 1911422
zbMATH Keywords: \(p\)-adic groups
zbMATH Keywords: Schur algebras
zbMATH Keywords: representations
zbMATH Keywords: affine quantum groups
MaRDI profile type: Publication
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full work available at URL: s0012-7094-03-11612-9
OpenAlex ID: W1580443957
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description: scientific article; zbMATH DE number 1911422