The automorphism equivariant Hitchin index

Let T be the one-dimensional complex torus. We consider the action of an automorphism f of a Riemann surface X on the cohomology of the T-equivariant determinant line bundle L over the moduli space M of rank two Higgs bundles on X with fixed determinant of odd degree. We define and study the automorphism equivariant Hitchin index χT(M,L,f). 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 X/〈f〉.