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. This relationship takes the form of identities between
orbital integrals on
reductive groupsG and H over a nonarchimedean
local fieldF, where the group H, called an
endoscopic group of G, is constructed from G and some additional data.
The first case considered was (
Labesse & Langlands 1979). Langlands and
Diana Shelstad (
1987) then developed the general framework for the theory of endoscopic transfer and formulated specific conjectures. However, during the next two decades only partial progress was made towards proving the fundamental lemma.[2][3] Harris called it a "bottleneck limiting progress on a host of arithmetic questions".[4] Langlands himself, writing on the origins of endoscopy, commented:
... it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of
Shimura varieties; it is the stabilized (or stable) trace formula, the reduction of the trace formula itself to the stable trace formula for a group and its endoscopic groups, and the stabilization of the
Grothendieck–Lefschetz formula. None of these are possible without the fundamental lemma and its absence rendered progress almost impossible for more than twenty years.[5]
Statement
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
F is a local field
G is an unramified group defined over F, in other words a quasi-split reductive group defined over F that splits over an unramified extension of F
H is an unramified endoscopic group of G associated to κ
KG and KH are hyperspecial maximal compact subgroups of G and H, which means roughly that they are the subgroups of points with coefficients in the ring of integers of F.
1KG and 1KH are the characteristic functions of KG and KH.
Δ(γH,γG) is a transfer factor, a certain elementary expression depending on γH and γG
γH and γG are elements of G and H representing stable conjugacy classes, such that the stable conjugacy class of G is the transfer of the stable conjugacy class of H.
κ is a character of the group of conjugacy classes in the stable conjugacy class of γG
SO and O are stable orbital integrals and orbital integrals depending on their parameters.
Approaches
Shelstad (1982) proved the fundamental lemma for Archimedean fields.
Waldspurger (1991) verified the fundamental lemma for general linear groups.
Hales (1997) and
Weissauer (2009) verified the fundamental lemma for the symplectic and general symplectic groups Sp4, GSp4.
A paper of
George Lusztig and
David Kazhdan pointed out that orbital integrals could be interpreted as counting points on certain algebraic varieties over finite fields. Further, the integrals in question can be computed in a way that depends only on the residue field of F; and the issue can be reduced to the Lie algebra version of the orbital integrals. Then the problem was restated in terms of the
Springer fiber of algebraic groups.[6] The circle of ideas was connected to a
purity conjecture; Laumon gave a conditional proof based on such a conjecture, for unitary groups. Laumon and Ngô (
2008) then proved the fundamental lemma for unitary groups, using
Hitchin fibration introduced by Ngô (
2006), which is an abstract geometric analogue of the
Hitchin system of complex algebraic geometry.
Waldspurger (2006) showed for Lie algebras that the function field case implies the fundamental lemma over all local fields, and
Waldspurger (2008) showed that the fundamental lemma for Lie algebras implies the fundamental lemma for groups.
Blasius, Don; Rogawski, Jonathan D. (1992), "Fundamental lemmas for U(3) and related groups", in Langlands, Robert P.; Ramakrishnan, Dinakar (eds.), The zeta functions of Picard modular surfaces, Montreal, QC: Univ. Montréal, pp. 363–394,
ISBN978-2-921120-08-1,
MR1155234
Kottwitz, Robert E. (1992), "Calculation of some orbital integrals", in Langlands, Robert P.; Ramakrishnan, Dinakar (eds.), The zeta functions of Picard modular surfaces, Montreal, QC: Univ. Montréal, pp. 349–362,
ISBN978-2-921120-08-1,
MR1155233
Langlands, Robert P. (1983),
Les débuts d'une formule des traces stable, Publications Mathématiques de l'Université Paris VII [Mathematical Publications of the University of Paris VII], vol. 13, Paris: Université de Paris VII U.E.R. de Mathématiques,
MR0697567