Cells in affine Weyl groups and tensor categories
type: Item
instance of: scholarly article
title: Cells in affine Weyl groups and tensor categories
zbMATH Open document ID: 0884.20026
published in: Advances in Mathematics
publication date: 1998-03-31T00:00:00Z
review text: We construct a tensor (or monoidal) category for any two-sided cell in a finite or affine Weyl group. The isomorphism classes of simple objects of this category correspond to the elements of the two-sided cell; the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold, which imitates geometrically the definition of the ring \(J\) given by \textit{G. Lusztig} [J. Algebra 109, 536-548 (1987; Zbl 0625.20032)]. Similarly, we construct, for any left cell, a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse (a right cell). This contains as a special case the category of equivariant perverse sheaves on an affine Grassmannian, which, as a consequence of \textit{G. Lusztig} [Astérisque 101/102, 208-229 (1983; Zbl 0561.22013)]\ is a tensor category for convolution (without truncation), closely related to the tensor category of representations of the group dual, in the sense of Langlands, to the group which gives rise to the affine Grassmannian. But our construction can be specialized in other ways, and it gives a (conjectural) realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the Langlands dual group.
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1042603
zbMATH Keywords: two-sided cells
zbMATH Keywords: affine Weyl groups
zbMATH Keywords: tensor products
zbMATH Keywords: left cells
zbMATH Keywords: tensor categories
zbMATH Keywords: categories of equivariant perverse sheaves
zbMATH Keywords: tensor categories of representations
author: G. Lusztig
MaRDI profile type: Publication
full work available at URL: aima.1997.1645
OpenAlex ID: W2060053899
cites work: Q3320507
cites work: Sheaves on a loop group and Langlands duality
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Q5185104
cites work: Character sheaves. II
cites work: Q3804915
cites work: Singularities of closures of K-orbits on flag manifolds
cites work: Canonical left cells in affine Weyl groups
DOI: 10.1006/AIMA.1997.1645
description: scientific article; zbMATH DE number 1042603