Weighted $L^2$–cohomology of Coxeter groups

Abstract

Given a Coxeter system $\left(W,S\right)$ and a positive real multiparameter $q$, we study the “weighted $L^2$–cohomology groups,” of a certain simplicial complex $\Sigma$ associated to $\left(W,S\right)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $\left(W,S\right)$ and the multiparameter $q$. They have a “von Neumann dimension” with respect to the associated “Hecke–von Neumann algebra” ${\mathcal{N}}_q$. The dimension of the $i$–th cohomology group is denoted $b_{q\left(\Sigma \right)}^i$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b_{q\left(\Sigma \right)}^i$ are the usual $L^2$–Betti numbers of buildings of type $\left(W,S\right)$ and thickness $q$. For a certain range of $q$, we calculate these cohomology groups as modules over ${\mathcal{N}}_q$ and obtain explicit formulas for the $b_{q\left(\Sigma \right)}^i$. The range of $q$ for which our calculations are valid depends on the region of convergence of the growth series of $W$. Within this range, we also prove a Decomposition Theorem for ${\mathcal{N}}_q$, analogous to a theorem of L Solomon on the decomposition of the group algebra of a finite Coxeter group.