  Consider a finite group $G$ acting on a Riemann surface $S$, and the
associated branched Galois cover $\pi_G:S \to Y=S/G$. We introduce the concept
of geometric signature for the action of $G$, and we show that it captures the
information of the geometric structure of the lattice of intermediate covers,
the information about the isotypical decomposition of the rational
representation of the group $G$ acting on the Jacobian variety $JS$ of $S$, and
the dimension of the subvarieties of the isogeny decomposition of $JS$. We also
give a version of Riemann's existence theorem, adjusted to the present setting.
