On the structure of Kazhdan-Lusztig cells for branched Dynkin diagrams
type: Item
instance of: scholarly article
title: On the structure of Kazhdan-Lusztig cells for branched Dynkin diagrams
zbMATH Open document ID: 0786.22024
DOI: 10.1016/0021-8693(92)90150-K
author: Devra Garfinkle
author: David A. jun. Vogan
published in: Journal of Algebra
publication date: 1993-03-28T00:00:00Z
review text: Because of the complexity of the Kazhdan-Lusztig conjectures, they are of limited value for calculation, and it is often valuable to have less information in a more explicit form. For instance, the \(\tau\)-invariant of \textit{W. Borho} and \textit{J. Jantzen} [Invent. Math. 39, 1-53 (1977; Zbl 0327.17002)] and \textit{M. Duflo} [Ann. Math., II. Ser. 105, 107-120 (1977; Zbl 0346.17011)] provides useful (but incomplete) information about the action of a simple reflection in a coherent continuation representation. Roughly, the information it provides comes from little \(SL(2)\)'s inside \(G\), the reductive Lie group under consideration. At the next level of sophistication is the generalized \(\tau\)-invariant, where one exploits some of the relations in a Weyl group arising from rank two Levi subgroups, and gets somewhat better information about coherent continuation representations [\textit{D. Vogan}, Duke Math. J. 46, 61-108 (1979; Zbl 0398.22021), Math. Ann. 242, 209-224 (1979; Zbl 0387.17007)]. It is natural to stop with rank two subgroups, since all the defining relations of a Weyl group appear at that level. However, coherent continuation representations have more structure than arbitrary Weyl group representations: one may expect the structure of coherent continuation representations to be restricted by the presence of a Levi factor of type \(D_ 4\) in ways that are difficult to predict with more elementary considerations. It is a purpose of the present paper to describe explicitly some restrictions on the nature of coherent continuation arising from Levi factor of type \(D_ 4\).
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 130474
zbMATH Keywords: character
zbMATH Keywords: coherent continuation representation
zbMATH Keywords: reductive Lie group
zbMATH Keywords: Weyl group
zbMATH Keywords: Levi factor of type \(D_ 4\)
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
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full work available at URL: 0021-8693(92)90150-k
OpenAlex ID: W2071419723
description: scientific article; zbMATH DE number 130474