The Real Chevalley Involution

We consider the Chevalley involution in the context of real reductive groups. We show that if G® is the real points of a connected reductive group, there is an involution, unique up to conjugacy by G®, taking any semisimple element to a conjugate of its inverse. As applications we give a condition for every irreducible representation of G® to be self-dual, and to the Frobenius Schur indicator for such groups.