Multiplicative Global Springer Theory

The moduli of Higgs bundles and the Hitchin fibration are central to many thriving research areas, such as mirror symmetry, non-abelian Hodge theory and the geometric Langlands program. They are also useful for globalizing representation-theoretic phenomena such as Springer theory and the study of Springer fibers. In the local and global settings, Springer fibers are replaced by affine Springer fibers and Hitchin fibers respectively, which parametrize Higgs bundles over a formal disc or over an algebraic curve. In 2011, Z. Yun globalized Springer theory by constructing an action of the extended affine Weyl group on the cohomology of parabolic Hitchin fibers. Meanwhile, there is an ongoing program to replicate the theory of Higgs bundles for the multiplicative case. This involves the study of multiplicative affine Springer fibers, the mutliplicative Hitchin fibration and their applications to the Fundamental Lemma. This thesis is a continuation of this program and it develops the theory of parabolic multiplicative affine Springer fibers and the parabolic multiplicative Hitchin fibration. With these constructions at hand, we are able to develop multiplicative global Springer theory.