The motive of the moduli stack of G-bundles over the universal curve

An abelian category of relative pure motives is constructed along the lines of Andr\'e (over a field of characteristic 0). An algebraic stack is shown to possess a motive in this sense. This motive is studied for the moduli stack of G-bundles over the universal curve for appropriate G, and a motivic Atiyah-Bott formula is obtained. By taking Hodge realization, a description of the corresponding variations of Hodge structures over M_g is given. In particular, it follows that these lie in the tensor category generated by the variation of Hodge structure associated to the curve itself. Similar results are also given for low degree variations of Hodge structures associated to the moduli space of stable G-bundles.