Semisimple Zariski closure of Coxeter groups

Abstract

Let W be an irreducible, finitely generated Coxeter group. The geometric representation provides an discrete embedding in the orthogonal group of the so-called Tits form. One can look at the representation modulo the kernel of this form; we give a new proof of the following result of Vinberg: if W is non-affine, then this representation remains faithful. Our proof uses relative Kazhdan Property (T). The following corollary was only known to hold when the Tits form is non-degenerate: the reduced C*-algebra of W is simple with a unique normalized trace. Some other corollaries are pointed out.