Multiplicative Hitchin Fibration and Fundamental Lemma

In this thesis, we thoroughly study the multiplicative Hitchin fibration, which is the group analogue of the usual Hitchin fibration by replacing the Lie algebra with a reductive monoid. As an application, we use it to prove the standard endoscopic fundamental lemma for adjoint groups. Although similar to its Lie algebra counterpart in many ways, the multiplicative Hitchin fibration has a lot more new features and is much more complicated. There are three main highlights in this thesis: the geometrization of endoscopic transfer, including the construction of endoscopic monoids; a local model of singularity that connects with representations of the dual group; a generalized support theorem which not only is the key to prove fundamental lemma but also reveals some potential new phenomenon that is not explained by endoscopy.