The moduli space of Higgs bundles admits a useful and important fibration, which is induced by taking the adjoint quotient. Thanks to Ngo's Product Formula in 2010, the (global) fibers can be expressed as a product of (local) affine Springer fibers, which not only parametrizes Higgs bundles locally, but also evaluates certain orbital integrals in the Fundamental Lemma. Meanwhile in 2011, Yun used the Hitchin fibration to develop global Springer theory by constructing actions of the extended affine Weyl group on the cohomology of Hitchin fibers. Since 2015, strides have been made by Bouthier, Chi and Wang to develop the group-theoretic or multiplicative analogue. Here, usual (linear) Higgs bundles are replaced by Higgs-Vinberg bundles while affine Springer fibers are replaced by generalized ones (a.k.a Kottwitz-Viehmann varieties). Since the latter can be seen as non-p-adic versions of affine Deligne Lusztig varieties, Chi used techniques from p-adic geometry in tandem with the global geometry to establish the non-emptiness pattern, equidimensionality and dimension formulae. My research seeks to generalize Yun's works and develop multiplicative global Springer theory. In this talk, I will recount past efforts, my current progress and future research directions.