Group Actions on Riemann-Roch Space

Let $\; G \;$ be a group acting on a compact Riemann surface $\; {\mathcal X} \;$ and $\; D \;$ be a $\; G$-invariant divisor on $\; {\mathcal X}. \;$ The action of $\; G \;$ on $\; {\mathcal X} \;$ induces a linear representation $\; L_G(D) \;$ of $\; G \;$ on the Riemann-Roch space associated to $\; D.$ In this paper we give some results on the decomposition of $\; L_G(D) \;$ as sum of complex irreducible representations of $\; G, \;$ for $\; D \;$ an effective non-special $\; G$-invariant divisor. In particular, we give explicit formulae for the multiplicity of each complex irreducible factor in $\; L_G(D) \;$. We work out some examples on well known families of curves.