Hyperbolic configurations of roots and Hecke algebras
type: Item
instance of: scholarly article
title: Hyperbolic configurations of roots and Hecke algebras
zbMATH Open document ID: 1201.20036
author: Jean Lécureux
published in: Journal of Algebra
publication date: 2010-04-21T00:00:00Z
review text: Let \(W\) be an infinite Coxeter group. If \(W\) is irreducible and non-affine, then it is shown in this paper by a geometric proof that for every root there are two roots and so that these three roots are pairwise different. The statement was shown combinatorially by Caprace and Rémy. Moreover, given a locally finite building \(X\) with Weyl group \(W\), and let \(G\) be a closed subgroup of the automorphism group of \(X\). Suppose that \(G\) is transitive on the pairs \((C,A)\) where \(C\) is a chamber in an apartment \(A\). Then \(G\) is locally compact endowed with a Haar measure \(dg\). Suppose \(K\) is a maximal compact subgroup of \(G\). Let \(L(G,K)\) be the continuous compactly supported complex functions on \(G\) which are \(K\) invariant under multiplication from left and from right. The convolution product turns \(L(G,K)\) into an algebra. In the paper it is shown that then the algebra \(L(G,K)\) is not commutative.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22F50
zbMATH DE Number: 5696916
zbMATH Keywords: buildings
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Hecke algebras
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.12.017
OpenAlex ID: W2092411787
cites work: Buildings
cites work: An elementary account of Selberg's lemma
cites work: Q4720067
cites work: Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup
cites work: Groupes et algébras de Lie
cites work: Factor and normal subgroup theorems for lattices in products of groups
cites work: Reductive groups over a local field
cites work: Q3885948
cites work: Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups.
cites work: Combinatorial and group-theoretic compactifications of buildings
cites work: Simplicity and superrigidity of twin building lattices.
cites work: Q5431018
cites work: A note on subgroups generated by reflections in Coxeter groups
cites work: Every Coxeter group acts amenably on a compact space
cites work: Cohomology of buildings and of their automorphism groups.
cites work: The conjugacy problem for Coxeter groups.
cites work: Q4057087
cites work: On the first Betti number of a constant negatively curved manifold
cites work: Q4948723
cites work: Q4329607
cites work: Buildings and Hecke algebras.
cites work: IRREDUCIBLE COXETER GROUPS
cites work: Topological groups of Kac-Moody type, right-angled twinnings and their lattices.
cites work: Theory of spherical functions on reductive algebraic groups over p-adic fields
cites work: Free subgroups in linear groups
cites work: Uniqueness and presentation of Kac-Moody groups over fields
DOI: 10.1016/J.JALGEBRA.2009.12.017
description: scientific article; zbMATH DE number 5696916