Frobenius-Schur indicators for subgroups and the Drinfel’d double of Weyl groups

If G is any 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 Cw(E) are defined over Q.