Let X be a compact Riemann surface and G = Aut(X) be its group of automorphisms (biholomorphisms between X and X). We know G acts on the space of harmonic forms and also on the space of holomorphic forms.

We know that the space of harmonic forms is a direct sum of the space of holomorphic forms and its conjugate.

Now, if we know the representation of G on the harmonic forms (by this I mean we have a matrix for each element of G), how can we find matrices for the representation of G on the holomorphic forms?