On the generic fibers and true base of parabolic $\mathrm{SO}_{2n}$-Hitchin systems

In this paper, we confirm a physical conjecture regarding the parabolic $\mathrm{SO}_{2n}$-Hitchin system, showing that Hitchin map factors through a finite cover of the Hitchin base that is isomorphic to an affine space. We first show that the generic Hitchin fiber is disconnected, with the number of components determined by the degree of the generalized Springer map, and then construct the cover explicitly. To this end, we introduce and study a new class of moduli spaces, termed \emph{residually nilpotent Hitchin systems}, and analyze their generic Hitchin fibers. Furthermore, we uncover an interesting connection between self-duality of the generic Hitchin fiber and special nilpotent orbits.