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In the mathematical theory of
automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...
s, the fundamental lemma relates orbital integrals on a
reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
over a
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
to stable orbital integrals on its
endoscopic group In mathematics, endoscopic groups of reductive algebraic groups were introduced by in his work on the stable trace formula. Roughly speaking, an endoscopic group ''H'' of ''G'' is a quasi-split group whose L-group is the connected component of ...
s. It was conjectured by in the course of developing the
Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
. The fundamental lemma was proved by
Gérard Laumon Gérard Laumon (; born 1952) is a French mathematician, best known for his results in number theory, for which he was awarded the Clay Research Award. Life and work Laumon studied at the École Normale Supérieure and Paris-Sud 11 University, Or ...
and
Ngô Bảo Châu Ngô Bảo Châu (, born June 28, 1972) is a Vietnamese-French mathematician at the University of Chicago, best known for proving the fundamental lemma for automorphic forms (proposed by Robert Langlands and Diana Shelstad). He is the first ...
in the case of
unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
s and then by for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of
Lie algebras In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
. ''
Time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
'' magazine placed Ngô's proof on the list of the "Top 10 scientific discoveries of 2009". In 2010, Ngô was awarded the
Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
for this proof.


Motivation and history

Langlands outlined a strategy for proving local and global
Langlands conjectures In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic nu ...
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 integral In mathematics, an orbital integral is an integral transform that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized spheres: for a homogeneous space ''X'' =&nb ...
s on
reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
s ''G'' and ''H'' over a nonarchimedean
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
''F'', where the group ''H'', called an
endoscopic group In mathematics, endoscopic groups of reductive algebraic groups were introduced by in his work on the stable trace formula. Roughly speaking, an endoscopic group ''H'' of ''G'' is a quasi-split group whose L-group is the connected component of ...
of ''G'', is constructed from ''G'' and some additional data. The first case considered was G = _2 . 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. Harris called it a "bottleneck limiting progress on a host of arithmetic questions". Langlands himself, writing on the origins of endoscopy, commented:


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 : :SO_(1_) = \Delta(\gamma_H,\gamma_G)O^\kappa_(1_) 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''K''''G'' and 1''K''''H'' are the characteristic functions of ''K''''G'' and ''K''''H''. *Δ(γ''H''''G'') is a transfer factor, a certain elementary expression depending on γ''H'' and γ''G''''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.


Approaches

proved the fundamental lemma for Archimedean fields. verified the fundamental lemma for general linear groups. and verified some cases of the fundamental lemma for 3-dimensional unitary groups. and verified the fundamental lemma for the symplectic and general symplectic groups Sp4, GSp4. A paper of
George Lusztig George Lusztig (born ''Gheorghe Lusztig''; May 20, 1946) is an American-Romanian mathematician and Abdun Nur Professor at the Massachusetts Institute of Technology (MIT). He was a Norbert Wiener Professor in the Department of Mathematics from ...
and
David Kazhdan David Kazhdan ( he, דוד קשדן), born Dmitry Aleksandrovich Kazhdan (russian: Дми́трий Александро́вич Кажда́н), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 Ma ...
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.The Fundamental Lemma for Unitary Groups
, at p. 12., Gérard Laumon The circle of ideas was connected to a
purity conjecture In the mathematical field of algebraic geometry, purity is a theme covering a number of results and conjectures, which collectively address the question of proving that "when something happens, it happens in a particular codimension". Purity of th ...
; Laumon gave a conditional proof based on such a conjecture, for unitary groups. then proved the fundamental lemma for unitary groups, using
Hitchin fibration In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the ...
introduced by , which is an abstract geometric analogue of the
Hitchin system In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the ...
of complex algebraic geometry. showed for Lie algebras that the function field case implies the fundamental lemma over all local fields, and showed that the fundamental lemma for Lie algebras implies the fundamental lemma for groups.


Notes


References

* * * * * * * * * * * * * * * * * * * *


External links


Gerard Laumon lecture on the fundamental lemma for unitary groups
* {{cite journal , last=Basken , first=Paul , url=http://chronicle.com/article/Understanding-the-Langlands/124368/ , title=Understanding the Langlands Fundamental Lemma , journal=
The Chronicle of Higher Education ''The Chronicle of Higher Education'' is a newspaper and website that presents news, information, and jobs for college and university faculty and student affairs professionals (staff members and administrators). A subscription is required to re ...
, date=September 12, 2010 Algebraic groups Automorphic forms Theorems in abstract algebra Lemmas in number theory Langlands program