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A relative Riemann-Hurwitz theorem, the Hurwitz-Hodge bundle, and orbifold Gromov-Witten theory

We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible Gcovers. This formula can be interpreted as a representation-ring-valued relative Riemann-Hurwitz formula for families of admissible G-covers. This formula yields an explicit description, without reference to G-covers, of the virtual class for orbifold Gromov-Witten invariants of a global quotient in degree zero. It also yields a new differential equation which computes arbitrary descendant Hurwitz-Hodge integrals. For G = Z2 and genus zero, we obtain HurwitzHodge integrals due to Faber-Pandharipande. We also calculate some HurwitzHodge integrals for G = Z3. In particular, we calculate some Gromov-Witten invariants of [C/Z3] and show agreement with the predictions from mirror symmetry due to Aganagic-Bouchard-Klemm in genus zero and one.