Big monodromy for higher Prym representations

Let ∑_g′ → ∑_g be a cover of an orientable surface of genus g by an orientable surface of genus g^′, branched at n points, with Galois group H. Such a cover induces a virtual action of the mapping class group Mod_g,n+1 of a genus g surface with n + 1 marked points on H¹.∑_g′, C). When g is large in terms of the group H, we calculate precisely the connected monodromy group of this action. The methods are Hodge-theoretic and rely on a “generic Torelli theorem with coefficients”.