Fundamental lemma (Langlands program)

In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was conjectured by Langlands (1983) in the course of developing the Langlands program. The fundamental lemma was proved by Gérard Laumon and Ngô Bảo Châu in the case of unitary groups and then by Ngô for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of Lie algebras. Time magazine placed Ngô's proof on the list of the "Top 10 scientific discoveries of 2009".^[1] In 2010 Ngô was awarded the Fields medal for this proof.

Motivation and history

Robert 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 groups G and H over a nonarchimedean local field F, where the group H, called an endoscopic group of G, is constructed from G and some additional data.

The first case considered was G = SL₂ (Labesse & Langlands 1979). Langlands and 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):

$SO_{\gamma_H}(1_{K_H}) = \Delta(\gamma_H,\gamma_G)O^\kappa_{\gamma_G}(1_{K_G})$

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
• K_G and K_H 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.
• 1_KG and 1_KH are the characteristic functions of K_G and K_H.
• Δ(γ_H,γ_G) is a transfer factor, a certain elementary expression depending on γ_H and γ_G
• γ_H and γ_G 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 γ_G
• SO and O are stable orbital integrals and orbital integrals depending on their parameters.If we have an orbital integral for a reductive group on a Calabi-Yau surface, then it is dual to a stable orbital integral for an endoscopic group on its mirror surface.

Approaches

Shelstad (1982) proved the fundamental lemma for Archimedean fields.

Waldspurger (1991) verified the fundamental lemma for general linear groups.

Kottwitz (1992) and Blasius & Rogawski (1992) verified some cases of the fundamental lemma for 3-dimensional unitary 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.

Notes

1. Top 10 Scientific Discoveries of 2009, Time
2. Kottwitz and Rogawski for U₃, Wadspurger for SL_n, Hales and Weissauer for Sp₄.
3. Fundamental Lemma and Hitchin Fibration, Gérard Laumon, May 13, 2009
4. INTRODUCTION TO “THE STABLE TRACE FORMULA, SHIMURA VARIETIES, AND ARITHMETIC APPLICATIONS”, p. 1., Michael Harris
5. publications.ias.edu
6. The Fundamental Lemma for Unitary Groups, at p. 12., Gérard Laumon

References

• Blasius, Don; Rogawski, Jonathan D. (1992), "Fundamental lemmas for U(3) and related groups", in Langlands, Robert P.; Ramakrishnan, Dinakar, The zeta functions of Picard modular surfaces, Montreal, QC: Univ. Montréal, pp. 363–394, ISBN 978-2-921120-08-1, MR 1155234
• Casselman, W. (2009), Langlands' Fundamental Lemma for SL(2) (PDF)
• Dat, Jean-François (November 2004), Lemme fondamental et endoscopie, une approche géométrique, d'après Gérard Laumon et Ngô Bao Châu (PDF), Séminaire Bourbaki, no 940
• Hales, Thomas C. (1997), "The fundamental lemma for Sp(4)", Proceedings of the American Mathematical Society 125 (1): 301–308, doi:10.1090/S0002-9939-97-03546-6, ISSN 0002-9939, MR 1346977
• Harris, M. (ed.), Stabilisation de la formule des traces, variétés de Shimura, et applications arithmétiques
• Kazhdan, David; Lusztig, George (1988), "Fixed point varieties on affine flag manifolds", Israel Journal of Mathematics 62 (2): 129–168, doi:10.1007/BF02787119, ISSN 0021-2172, MR 947819
• Kottwitz, Robert E. (1992), "Calculation of some orbital integrals", in Langlands, Robert P.; Ramakrishnan, Dinakar, The zeta functions of Picard modular surfaces, Montreal, QC: Univ. Montréal, pp. 349–362, ISBN 978-2-921120-08-1, MR 1155233
• Labesse, Jean-Pierre; Langlands, R. P. (1979), "L-indistinguishability for SL(2)", Canadian Journal of Mathematics 31 (4): 726–785, doi:10.4153/CJM-1979-070-3, ISSN 0008-414X, MR 540902
• 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] 13, Paris: Université de Paris VII U.E.R. de Mathématiques, MR 697567
• Langlands, Robert P.; Shelstad, Diana (1987), "On the definition of transfer factors", Mathematische Annalen 278 (1): 219–271, doi:10.1007/BF01458070, ISSN 0025-5831, MR 909227
• Laumon, Gérard (2006), "Aspects géométriques du Lemme Fondamental de Langlands-Shelstad", International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 401–419, MR 2275603
• Laumon, Gérard; Ngô, Bao Châu (2008), "Le lemme fondamental pour les groupes unitaires", Annals of Mathematics. Second Series 168 (2): 477–573, doi:10.4007/annals.2008.168.477, ISSN 0003-486X, MR 2434884
• Nadler, David (2012), "The geometric nature of the fundamental lemma", Bulletin of the American Mathematical Society 49: 1–50, doi:10.1090/S0273-0979-2011-01342-8, ISSN 0002-9904
• Ngô, Bao Châu (2006), "Fibration de Hitchin et endoscopie", Inventiones Mathematicae 164 (2): 399–453, doi:10.1007/s00222-005-0483-7, ISSN 0020-9910, MR 2218781
• Ngô, Bao Châu (2010), "Le lemme fondamental pour les algèbres de Lie", Institut des Hautes Études Scientifiques. Publications Mathématiques 111: 1–169, doi:10.1007/s10240-010-0026-7, ISSN 0073-8301, MR 2653248
• Shelstad, Diana (1982), "L-indistinguishability for real groups", Mathematische Annalen 259 (3): 385–430, doi:10.1007/BF01456950, ISSN 0025-5831, MR 661206
• Waldspurger, Jean-Loup (1991), "Sur les intégrales orbitales tordues pour les groupes linéaires: un lemme fondamental", Canadian Journal of Mathematics 43 (4): 852–896, doi:10.4153/CJM-1991-049-5, ISSN 0008-414X, MR 1127034
• Waldspurger, Jean-Loup (2006), "Endoscopie et changement de caractéristique", Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu 5 (3): 423–525, doi:10.1017/S1474748006000041, ISSN 1474-7480, MR 2241929
• Waldspurger, Jean-Loup (2008), "L'endoscopie tordue n'est pas si tordue" [Twisted endoscopy is not so twisted] (PDF), Memoirs of the American Mathematical Society (in French) (Providence, R.I.: American Mathematical Society) 194 (908): 261, ISBN 978-0-8218-4469-4, ISSN 0065-9266, MR 2418405
• Weissauer, Rainer (2009), Endoscopy for GSp(4) and the cohomology of Siegel modular threefolds, Lecture Notes in Mathematics 1968, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-89306-6, ISBN 978-3-540-89305-9, MR 2498783

External links

• Gerard Laumon lecture on the fundamental lemma for unitary groups
• Basken, Paul (September 12, 2010). "Understanding the Langlands Fundamental Lemma". The Chronicle of Higher Education.