  Let T be the one-dimensional complex torus. We consider the action of an
automorphism of a Riemann surface X on the cohomology of the T-equivariant
determinant line bundle over the moduli space of rank two Higgs bundles on X
with fixed determinant of odd degree. We define and study the automorphism
equivariant Hitchin index. We prove a formula for it in terms of cohomological
pairings of canonical T-equivariant classes of certain moduli spaces of
parabolic Higgs bundles over the quotient Riemann surface.
