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 Harm(X) of harmonic forms and also on the space Omega(X) of holomorphic forms.

We know that Harm(X) is a direct sum of Omega(X) and its conjugate.

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

ADDED: I am assuming we have no information about Omega(X) or Harm(X). We just know the genus of X, G (with a multiplication table) and the representation of G on Harm(X) (the matrices associated to each element of the group).