  The rational homology of unordered configuration spaces of points on any
surface was studied by Drummond-Cole and Knudsen. We compute the rational
cohomology of configuration spaces on a closed orientable surface, keeping
track of the mixed Hodge numbers and the action of the symplectic group on the
cohomology. We find a series with coefficients in the Grothendieck ring of
sp(2g) that describes explicitly the decomposition of the cohomology into
irreducible representations. From that we deduce the mixed Hodge numbers and
the Betti numbers, obtaining a new formula without cancellations.
