  If G is a finite group and k is a field, there is a natural construction of a
Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that
if G is any finite real reflection group with Drinfel'd double D(G) over an
algebraically closed field k of characteristic not 2, then every simple
D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical
results for modules over the group itself. We also prove some new results about
Weyl groups. In particular, we prove that any abelian subgroup is inverted by
some involution. Also, if E is any elementary abelian 2-subgroup of the Weyl
group W, then all representations of the centralizer of E in W are defined over
the rational field.
