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