A proof of the endoscopic fundamental lemma for spherical Hecke algebras using a new comprehensive framework of multiplicative Hitchin fibrations This book develops a comprehensive framework of multiplicative Hitchin fibrations, which is a novel analogue of the classical Hitchin fibrations, and then uses it to prove the endoscopic fundamental lemma for spherical Hecke algebras over local function fields. Unlike previous proofs of similar results, it does not rely on reduction to Lie algebras or harmonic analysis; rather, it features a direct geometric proof inspired by B. C. Ngô's monumental work in the Lie algebra case. To achieve this, the book carries out in-depth study of the invariant theory of very flat reductive monoids and discovers, through the induced local and global geometry, new connections with Langlands duality, endoscopic theory, and representation theory. One highlight among these is the local study of multiplicative Springer fibers, the intricate arithmetic properties of which lead to a surprising relation between endoscopic transfer and crystal bases via an asymptotic analogue of the fundamental lemma. Restrictions of the crystal bases, unlike representations, remain largely mysterious to this day, and the asymptotic fundamental lemma may be viewed as one such instance that is made concrete. On the other hand, its proof is through global geometry, which provides new clues about how to better approach representation-theoretic objects like the crystal bases.

Other highlights include the deformation of multiplicative Higgs bundles and a support theorem, which not only help establish a geometric form of endoscopic transfer but also demonstrate the potential of this method for future study of more general functoriality phenomena.