Weighted $L^2$-cohomology of Coxeter groups

Abstract

Given a Coxeter system $(W,S)$ and a positive real multiparameter $\bq$, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $Σ$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. They have a "von Neumann dimension" with respect to the associated "Hecke - von Neumann algebra," $N_q$. The dimension of the $i^th$ cohomology group is denoted $b^i_q(Σ)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(Σ)$ are the usual $L^2$-Betti numbers of buildings of type $(W,S)$ and thickness $q$. For a certain range of $q$, we calculate these cohomology groups as modules over $N_q$ and obtain explicit formulas for the $b^i_q(Σ)$. 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 $N_q$, analogous to a theorem of L. Solomon on the decomposition of the group algebra of a finite Coxeter group.