Universal module in characteristic \(l>0\) associated to a character of the Hecke algebra of \(\text{GL}(n)\) over a \(p\)-adic field with \(l\neq p\)
type: Item
instance of: scholarly article
title: Universal module in characteristic \(l>0\) associated to a character of the Hecke algebra of \(\text{GL}(n)\) over a \(p\)-adic field with \(l\neq p\)
zbMATH Open document ID: 0920.22010
author: Xavier Lazarus
published in: Journal of Algebra
publication date: 1999-09-09T00:00:00Z
review text: Let \(F\) be a non-archimedean local field with finite residue field and let \(G = \text{GL}(n,F)\). Let \(\chi\) be an unramified character of the diagonal subgroup of \(G\) and let \(\lambda\) denote the character of the Hecke algebra corresponding to \(\chi\) by the Satake map. We then have the principal series representation \(I_{\chi}\) of \(G\) and the universal module (or induced module) \(M_{\lambda}\) associated to \(\lambda\). Moreover, there is a natural map \(M_{\lambda} \rightarrow I_{\chi}\). This has been studied for characters with values in \(\mathbb C\) and representations over \(\mathbb C\). The present paper is concerned with the same situation with \(\mathbb C\) replaced by an algebraically closed field of characteristic \(l > 0\) and different from the residual characteristic of \(F\). It is proved that the \(G\)-modules \(M_{\lambda}\) and \(I_{\chi}\) have the same, finite, Jordan-Hölder sequence, under the assumption that \(l\) does not divide the number of elements of the reduced group \(\overline{G} = \text{GL}(n,\mathbb{F}_q)\). There is also a criterion for \(M_{\lambda} \rightarrow I_{\chi}\) to be an isomorphism, under the assumption that \(l\) does not divide \(q(q - 1)\). The main tool for the proofs is the functor \(V\mapsto V^B\) from representations of \(G\) to modules over the Hecke algebra of \(G\) with respect to an Iwahori subgroup \(B\). There are some inaccuracies in the text (e.g., \(\delta\) defined in 1.1 is the inverse of \(\delta\) used further on; Proposition 1.20 is not in concordance with the definition 1.8). Lemma 1.11(i) is false (the right hand side of the equality must be multiplied by an integral power of \(q\)).
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 1270869
zbMATH Keywords: character
zbMATH Keywords: Hecke algebra
zbMATH Keywords: principal series representation
zbMATH Keywords: universal module
zbMATH Keywords: Jordan-Hölder sequence
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: jabr.1998.7641
OpenAlex ID: W2036616423
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DOI: 10.1006/JABR.1998.7641
description: scientific article; zbMATH DE number 1270869