Weighted L–cohomology of Coxeter groups

Given a Coxeter system .W;S/ and a positive real multiparameter q , we study the “weighted L–cohomology groups,” of a certain simplicial complex † 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” Nq . The dimension of the i –th cohomology group is denoted b q.†/ . It is a nonnegative real number which varies continuously with q . When q is integral, the b q.†/ are the usual L –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 Nq and obtain explicit formulas for the b q.†/ . 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 Nq , analogous to a theorem of L Solomon on the decomposition of the group algebra of a finite Coxeter group.