Polarizations of Prym varieties for Weyl groups via abelianization

Let $\pi: Z \ra X$ be a Galois covering of smooth projective curves withGalois group the Weyl group of a simple and simply-connected Lie group $G$. Forany dominant weight $\lambda$ consider the curve $Y = Z/\Stab(\lambda)$. TheKanev correspondence defines an abelian subvariety $P_\lambda$ of the Jacobianof $Y$. We compute the type of the polarization of the restriction of thecanonical principal polarization of $\Jac(Y)$ to $P_\lambda$ in some cases. Inparticular, in the case of the group $E_8$ we obtain families of Prym-Tyurinvarieties. The main idea is the use of an abelianization map of the Donagi-Prymvariety to the moduli stack of principal $G$-bundles on the curve $X$.