  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.
