On spaces associated with invariant divisors on Galois covers of Riemann surfaces and their applications

Let f : X → S be a Galois cover of Riemann surfaces, with Galois group G. I. this paper we analyze the G-invariant divisors on X, and their associated spaces of meromorphic functions, differentials, and q-differentials. We generalize the trace formula for non-trivial elements of G on q-differentials, as well as the Chevalley–Weil Formula. When G is Abelian or when the genus of S is 0 we prove additional results, and we also determine the non-special G-invariant divisors when both conditions are satisfied.