AUTHORS
T. Terragni and T.S. Weigel

TITLE
The Coxeter complex and the Euler characteristic of a Hecke algebra

ABSTRACT
For any Hecke algebra $H=H_q(W,S)$ associated to a Coxeter group $(W,S)$ and a distinguished element $q\in R$ of a commutative ring with unit $R$ we introduce a finite chain complex of left $H$-modules $(C,\partial)$ which reflects many properties of the Coxeter complex of $(W,S)$, i.e., it is acyclic if $(W,S)$ is non-spherical (cf. Thm. A), and $H$ is of type FP under suitable conditions on the distinguished element $q\in R$ (cf. Prop. B). 
There exists a canonical trace function $\mu: H\to R$ (cf. Prop. 5.1). This trace function $\mu$ evaluated on the Hattori-Stallings rank of $(C,\partial)$ can be considered as the Euler characteristic $\chi_H$ of $H$. It will be shown that for generic values of $q$ the Euler characteristic coincides with the reciprocal of the Poincaré series of $(W,S)$ evaluated in $q$ (cf. Thm. C).

STATUS
Submitted for publication