de Rham and Dolbeault cohomology of solvmanifolds with local systems

Let G be a simply connected solvable Lie group with a lattice G and the Lie algebra g and a representation. : G -> GL(V-p) whose restriction on the nilradical is unipotent. Consider the flat bundle E-p given by p. By using "many" characters {alpha} of G and "many" flat line bundles {E-alpha} over G/G, we show that an isomorphism circle plus({alpha}) II*(g, V-alpha circle times V-p) congruent to circle plus({E alpha}) II* (G/Gamma, E-alpha circle times E-p) holds. This isomorphism is a generalization of the well-known fact: "If G is nilpotent and. is unipotent then, the isomorphism H*(g, V-p) congruent to H*(G/Gamma, E.) holds". By this result, we construct an explicit finite-dimensional cochain complex which compute the cohomology H*(G/Gamma, E.) of solvmanifolds even if the isomorphism H*(g, V-p) congruent to H*(G/Gamma, E.) does not hold. For Dolbeault cohomology of complex parallelizable solvmanifolds, we also prove an analogue of the above isomorphism result which is a generalization of computations of Dolbeault cohomology of complex parallelizable nilmanifolds. By this isomorphism, we construct an explicit finite-dimensional cochain complex which compute the Dolbeault cohomology of complex parallelizable solvmanifolds.