On the Kazhdan-Lusztig order on cells and families.
type: Item
instance of: scholarly article
title: On the Kazhdan-Lusztig order on cells and families.
zbMATH Open document ID: 1264.20005
DOI: 10.4171/CMH/273
author: Meinolf Geck
published in: Commentarii Mathematici Helvetici
publication date: 2012-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0910.4689
review text: Let \(\mathrm{Irr}(W)\) be the set of irreducible characters of a finite Coxeter group \(W\) over the complex number field \(\mathbb C\). Then \(\mathrm{Irr}(W)\) can be partitioned in two ways: one is in terms of Kazhdan-Lusztig two-sided cells of \(W\) (see \textit{D. Kazhdan} and \textit{G. Lusztig}, [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]), another is in terms of families [\textit{G. Lusztig}, Characters of reductive groups over a finite field. Ann. Math. Stud. 107. Princeton: Princeton University Press (1984; Zbl 0556.20033)]. These two partitions are the same. The theory of Kazhdan-Lusztig cells provides a natural partial order \(\leqslant_{LR}\) on the pieces of the partition. The main purpose of the present paper is to obtain a better understanding of the partial order \(\leqslant_{LR}\). The author shows that \(\leqslant_{LR}\) can be characterized in a purely elementary way in terms of standard operations (i.e., induction, truncated induction, tensoring with sign) in the character ring of \(W\). An efficient algorithm for computing the partial order can be implemented in CHEVIE, [\textit{M. Geck} et al., Appl. Algebra Eng. Commun. Comput. 7, No. 3, 175-210 (1996; Zbl 0847.20006)]. When \(W\) is the Weyl group of an algebraic group \(G\), the author shows that the partial order \(\leqslant_{LR}\) on the families of \(\mathrm{Irr}(W)\) can be interpreted via the Springer correspondence in terms of the closure relation among the special unipotent classes of \(G\). Note that a version of the latter result appeared in the paper by \textit{D. Barbasch} and \textit{D. A. Vogan}, [Ann. Math. (2) 121, 41-110 (1985; Zbl 0582.22007)]. The author's proof is based on the results of \textit{N. Spaltenstein}, [J. Reine Angew. Math. 343, 212-220 (1983; Zbl 0503.20012)], which is different from that of \textit{D. Barbasch} and \textit{D. A. Vogan} [loc. cit.].
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 6104830
zbMATH Keywords: finite Coxeter groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: Kazhdan-Lusztig cells
zbMATH Keywords: Springer correspondence
zbMATH Keywords: Weyl groups
zbMATH Keywords: algebraic groups
describes a project that uses: Coxeter
MaRDI profile type: Publication
arXiv ID: 0910.4689
description: scientific article