The G-generalized Hitchin systems and Prym varieties

Abstract

In this article, we consider the generalized Hitchin systems, introduced by Bottacin and Markman, for an arbitrary reductive complex group G; they have the structure of Lagrangian fibrations Pr→U by generalized Prym varieties over sets U parametrizing families of Weyl-invariant curves. The G-generalized Hitchin systems satisfy a rank two condition and one can find invariant varieties X which distinguish these systems. It is then shown that there is a correspondence between these integrable systems and the varieties X=KΣ[D]⊗h̃, which are equipped with an appropriate two form with values in the Cartan subalgebra 𝔥.