whose quotient curve has genus at least $3$, the natural map from ${{\mathbb Q}}G$ to the rational endomorphism algebra of the Jacobian of $C$ is an isomorphism (Corollary \[endo2\]). We also obtain a result of this kind in the hyperelliptic case (Theorem \[endo\] and Theorem \[theorem:GN\]).

Although we believe that this has an interest of its own, our motivation has been of a more topological nature, since it has implications for certain virtual linear representations of the (pointed) mapping class groups, as we shall now explain. Let $S$ be a closed oriented connected surface of genus $\ge 2$ endowed with an action of a finite group $G$ by orientation preserving diffeomorphisms. Then $G$ embeds in the mapping class group ${\operatorname{Mod}}(S)$ of $S$. The centralizer ${\operatorname{Mod}}(S)^G$ of $G$ in ${\operatorname{Mod}}(S)$ maps to a finite index subgroup of the mapping class group ${\operatorname{Mod}}(S_G)$ of the orbifold surface $S_G$ (these are mapping classes of the underlying surface which take orbifold points to orbifold points of the same type), with kernel the center $Z(G)$ of $G$. Thus ${\operatorname{Mod}}(S)^G$ and ${\operatorname{Mod}}(S_G)$ have in common a subgroup of finite index so that the natural representation of ${\operatorname{Mod}}(S)^G$ on $H^1(S,{{\mathbb Q}})$ can be regarded as a virtual 