Faithfulness of Actions on Riemann-Roch Spaces

Given a faithful action of a finite groupGon an algebraic curveXof genusgX≥ 2, we giveexplicit criteria for the induced action ofGon the Riemann–Roch spaceH0(X,OX(D)) to be faithful,whereDis aG-invariant divisor on X of degree at least 2gX− 2. This leads to a concise answer to the question of when the action ofGon the spaceH0(X,Ωx⊗m) of global holomorphic polydifferentials of order m is faithful. IfXis hyperelliptic, we provide an explicit basis of H0(X,Ωx⊗m). Finally, we giveapplications in deformation theory and in coding theory and discuss the analogous problem for theaction ofGon the first homologyH1(X,ℤ/mℤ) ifXis a Riemann surface.