Polarizations of Prym Varieties for Weyl Groups via Abelianization

Let π : Z → X be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group G. For any dominant weight λ consider the curve Y = Z/Stab(λ). The Kanev correspondence defines an abelian subvariety Pλ of the Jacobian of Y . We compute the type of the polarization of the restriction of the canonical principal polarization of Jac(Y ) to Pλ in some cases. In particular, in the case of the group E8 we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi-Prym variety to the moduli stack of principal G-bundles on the curve X .