The based ring of the lowest two-sided cell of an affine Weyl group
type: Item
instance of: scholarly article
title: The based ring of the lowest two-sided cell of an affine Weyl group
zbMATH Open document ID: 0709.20021
DOI: 10.1016/0021-8693(90)90059-W
author: Nanhua Xi
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let G be a simply connected, almost simple complex algebraic group and T a maximal torus of G. For the root lattice \(P\subset X=Hom(T,{\mathbb{C}}^*)\) and the Weyl group \(W_ 0=N_ G(T)/T\) of G, the semidirect product \(W=W_ 0\ltimes P\) is the affine Weyl group associated to G. Let \(J_ 0\) be the based ring associated to the lowest two-sided cell of W (see [\textit{G. Lusztig}, Proc. Symp. Pure Math. 47, 235-262 (1987; Zbl 0657.20037)] for the definition of a based ring). For a G-set Y, let \(K_ G(Y\times Y)\) be the Grothendieck group of the category of G-equivariant complex vector bundles on the G-set \(Y\times Y\). \(K_ G(Y\times Y)\) has a based ring structure if we define the multiplication * by setting \[ (V*U)_{(x,y)}=\oplus_{z\in Y}V_{(x,z)}\otimes U_{(z,y)} \] for any \(V,U\in K_ G(Y\times Y)\) and define \(\tilde V_{(x,y)}=V^*_{(y,x)}\) (the complex dual of \(V_{(y,x)})\). Then the main result of this paper is to assert that \(J_ 0\) is isomorphic as a based ring to \(K_ G(Y\times Y)\) for a certain G-set Y with \(| Y| =| W_ 0|\). This result supports a conjecture of \textit{G. Lusztig} [see J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No.2, 297-328 (1989; Zbl 0688.20020)].
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 4165168
zbMATH Keywords: simply connected, almost simple complex algebraic group
zbMATH Keywords: maximal torus
zbMATH Keywords: root lattice
zbMATH Keywords: affine Weyl group
zbMATH Keywords: based ring
zbMATH Keywords: two-sided cell
zbMATH Keywords: Grothendieck group
zbMATH Keywords: category of G-equivariant complex vector bundles
zbMATH Keywords: G-set
MaRDI profile type: Publication
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Hecke algebras and Jantzen's generic decomposition patterns
cites work: Q5185104
cites work: Distinguished conjugacy classes and elliptic Weyl group elements.
cites work: Canonical left cells in affine Weyl groups
cites work: A Two-Sided Cell in an Affine Weyl Group, II
description: scientific article; zbMATH DE number 4165168