  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.
