Higgs Bundles with Nilpotent Structures

We will talk various Hitchin systems with nilpotent structures at marked points. We will start with a relation between singularities of spectral curves and Kazhdan–Lusztig maps for nilpotent orbits. Then we will show a relation between duality of generic fibers and Springer correspondence for nilpotent orbits for Langlands dual groups ${\rm Sp}_{2n}$ and ${\rm SO}_{2n+1}$.