The fundamental lemma was proved by Gérard Laumon and Ngô Bảo Châu in the case of unitary groups and then by Ngô (2010) for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of Lie algebras.
Langlands outlined a strategy for proving local and global Langlands conjectures using the Arthur–Selberg trace formula, but in order for this approach to work, the geometric sides of the trace formula for different groups must be related in a particular way.
Langlands and Diana Shelstad (1987) then developed the general framework for the theory of endoscopic transfer and formulated specific conjectures.
[5]The fundamental lemma states that an orbital integral O for a group G is equal to a stable orbital integral SO for an endoscopic group H, up to a transfer factor Δ (Nadler 2012): where Shelstad (1982) proved the fundamental lemma for Archimedean fields.
Kottwitz (1992) and Blasius & Rogawski (1992) verified some cases of the fundamental lemma for 3-dimensional unitary groups.