Tautological classes with twisted coefficients

Let Mg be the moduli space of smooth genus g curves. We define a notion of Chow groups of Mg with coefficients in a representation of Sp(2g), and we define a subgroup of tautological classes in these Chow groups with twisted coefficients. Studying the tautological groups of Mg with twisted coefficients is equivalent to studying the tautological rings of all fibered powers Cn g of the universal curve Cg → Mg simultaneously. By taking the direct sum over all irreducible representations of the symplectic group in fixed genus, one obtains the structure of a twisted commutative algebra on the tautological classes. We obtain some structural results for this twisted commutative algebra, and we are able to calculate it explicitly when g ≤ 4. Thus we completely determine the tautological rings of all fibered powers of the universal curve over Mg in these genera. We also give some applications to the Faber conjecture.