Hyperbolic configurations of roots and Hecke algebras
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.
scientific article; zbMATH DE number 5696916
scientific article