Let \( \pi: Z \ra X \) be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group . For any dominant weight consider the curve \( Y = Z/\Stab(\lambda) \). The Kanev correspondence defines an abelian subvariety of the Jacobian of . We compute the type of the polarization of the restriction of the canonical principal polarization of \( \Jac(Y) \) to in some cases. In particular, in the case of the group 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 -bundles on the curve .
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Christoph Scheven, Christian Pauly, Polarizations of Prym varieties for Weyl groups via abelianization. J. Eur. Math. Soc. 11 (2009), no. 2, pp. 315–349DOI 10.4171/JEMS/152