Duality of Hitchin fibrations and endoscopy

The Hitchin fibration is a completely integral algebraic system thatfirst appeared in physical mathematics. This system with its particularly rich geometry emerged as a central object in the geometric Langlands program, in work by Beilinson-Drinfeld, Laumon, Witten, in particular through a remarkable duality between Hitchin fibrations of dual groups in the Langlands sense, discovered by Hausel-Thaddeus and Donagi-Pantev. He also played a central role in proving the fundamental Langlands-Shelstad lemma in the Langlands arithmetic program. This first demonstration is the combination of a fine-tuned study of the geometry of the Hitchin fibration inspired by automorphic endoscopy theory and that of the Beilinson-Bernstein-Deligne decomposition theorem in this particular framework. Recently, Groechenig, Wyss and Ziegler have proposed a new proof of the fundamental lemma, also based on the geometry of the Hitchin fibration, but instead of the difficult study of perverse beams in the Beilinson-Bernstein-Deligne decomposition, they use p-adic integration. Beyond this technical innovation, the duality of Hitchin fibrations plays a major role in this new proof. My lecture will cover these developments, focusing on the phenomenon of duality in Hitchin fibrations.