Local Langlands conjectures

In mathematics, the local Langlands conjectures, introduced by Langlands (1967, 1970), are part of the Langlands program. They describe a correspondence between representations of the Weil group of a local field and representations of algebraic groups over the local field, generalizing local class field theory from abelian Galois groups to non-abelian Galois groups.

1 Local Langlands conjectures for GL₁
2 Representations of the Weil group
3 Representations of GL_n(F)
4 Local Langlands conjectures for GL₂
5 Local Langlands conjectures for GL_n
6 Local Langlands conjectures for other groups
7 References
8 External links

Local Langlands conjectures for GL₁

The local Langlands conjectures for GL₁(K) follow from (and are essentially equivalent to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL₁(K)= K^* to the abelianization of the Weil group. In particular irreducible smooth representations of GL₁(K) are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL₁(C). This gives the Langlands correspondence between homomorphisms of the Weil group to GL₁(C) and irreducible smooth representations of GL₁(K).

Representations of the Weil group

Representations of the Weil group do not quite correspond to irreducible smooth representations of general linear groups. To get a bijection, one has to slightly modify the notion of a representation of the Weil group, to something called a Weil–Deligne representation. This consists of a representation of the Weil group on a vector space V together with a nilpotent endomorphism N of V such that wNw⁻¹=||w||N, or equivalently a representation of the Weil–Deligne group. In addition the representation of the Weil group should have an open kernel, and should be (Frobenius) semisimple.

For every Frobenius semisimple complex n-dimensional Weil–Deligne representations ρ of the Weil group of F there is an L-function L(s,ρ) and a local ε-factor ε(s,ρ,ψ) (depending on a character ψ of F).

Representations of GL_n(F)

The representations of GL_n(F) appearing in the local Langlands correspondence are smooth irreducible complex representations.

"Smooth" means that every vector is fixed by some open subgroup.
"Irreducible" means that the representation is nonzero and has no subrepresentations other than 0 and itself.

Smooth irreducible representations are automatically admissible.

The Bernstein–Zelevinsky classification reduces the classification of irreducible smooth representations to cuspidal representations.

For every irreducible admissible complex representation π there is an L-function L(s,π) and a local ε-factor ε(s,π,ψ) (depending on a character ψ of F). More generally, if there are two irreducible admissible representations π and π' of general linear groups there are local Rankin–Selberg convolution L-functions L(s,π×π') and ε-factors ε(s,π×π',ψ).

Bushnell & Kutzko (1993) described the irreducible admissible representations of general linear groups over local fields.

Local Langlands conjectures for GL₂

The local Langlands conjecture for GL₂ of a local field says that there is a (unique) bijection π from 2-dimensional semisimple Deligne representations of the Weil group to irreducible smooth representations of GL₂(F) that preserves L-functions, ε-factors, and commutes with twisting by characters of F^*.

Jacquet & Langlands (1970) verified the local Langlands conjectures for GL₂ in the case when the residue field does not have characteristic 2. In this case the representations of the Weil group are all of cyclic or dihedral type. Gelfand & Graev (1962) classified the smooth irreducible representations of GL₂(F) when F has odd residue characteristic (see also (Gelfand, Graev & Pyatetskii-Shapiro 1969, chapter 2)), and claimed incorrectly that the classification for even residue characteristic differs only insignifictanly from the odd residue characteristic case. Weil (1974) pointed out that when the residue field has characteristic 2, there are some extra exceptional 2-dimensional representations of the Weil group whose image in PGL₂(C) is of tetrahedral or octahedral type. (For global Langlands conjectures, 2-dimensional representations can also be of icosaheral type, but this cannot happen in the local case as the Galois groups are solvable.) Tunnell (1978) proved the local Langlands conjectures for the general linear group GL₂(K) over the 2-adic numbers, and over local fields containing a cube root of unity. Kutzko (1980, 1980b) proved the local Langlands conjectures for the general linear group GL₂(K) over all local fields.

Cartier (1981) and Bushnell & Henniart (2006) gave expositions of the proof.

Local Langlands conjectures for GL_n

The local Langlands conjectures for general linear groups state that there are unique bijections π ↔ ρ_π from equivalence classes of irreducible admissible representations π of GL_n(F) to equivalence classes of continuous Frobenius semisimple complex n-dimensional Weil–Deligne representations ρ_π of the Weil group of F, that preserve L-functions and ε-factors of pairs of representations, and coincide with the Artin map for 1-dimensional representations. In other words,

L(s,ρ_π⊗ρ_π') = L(s,π×π')
ε(s,ρ_π⊗ρ_π',ψ) = ε(s,π×π',ψ)

Laumon, Rapoport & Stuhler (1993) proved the local Langlands conjectures for the general linear group GL_n(K) for positive characteristic local fields K. Carayol (1992) gave an exposition of their work.

Richard Taylor and Michael Harris (2001) proved the local Langlands conjectures for the general linear group GL_n(K) for characteristic 0 local fields K. Henniart (2001) gave another proof. Carayol (2000) and Wedhorn (2008) gave expositions of their work.

Local Langlands conjectures for other groups

Borel (1979) and Vogan (1993) discuss the Langlands conjectures for more general groups. As of 2011, the Langlands conjectures for arbitrary reductive groups G are not as precise as the ones for general linear groups, and it is unclear what the correct way of stating them should be. Roughly speaking, admissible representations of a reductive group are grouped into disjoint finite sets called L-packets, which should correspond to some classes of homomorphisms, called L-parameters, from the Weil–Deligne group to the L-group of G.

Langlands (1989) proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible admissible representations (up to infinitesimal equivalence), or, equivalently, of their irreducible $(\mathfrak{g},K)$ -modules.

Gan & Takeda (2011) proved the local Langlands conjectures for the symplectic similitude group GSp(4) and used that in Gan & Takeda (2010) to deduce it for the symplectic group Sp(4).

References

Borel, Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W., Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, ISBN 978-0-8218-1437-6, MR 546608, http://www.ams.org/publications/online-books/pspum332-index
Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120
Bushnell, Colin J.; Kutzko, Philip C. (1993), The admissible dual of GL(N) via compact open subgroups, Annals of Mathematics Studies, 129, Princeton University Press, ISBN 978-0-691-03256-6; 978-0-691-02114-0, MR 1204652, http://books.google.com/books?id=J3bciC6TYY8C
Carayol, Henri (1992), "Variétés de Drinfeld compactes, d'après Laumon, Rapoport et Stuhler", Astérisque 206: 369–409, ISSN 0303-1179, MR 1206074, http://www.numdam.org/item?id=SB_1991-1992__34__369_0
Carayol, Henri (2000), Preuve de la conjecture de Langlands locale pour GL_n: travaux de Harris-Taylor et Henniart, "Séminaire Bourbaki. Vol. 1998/99.", Astérisque 266: 191–243, ISSN 0303-1179, MR 1772675, http://www.numdam.org/item?id=SB_1998-1999__41__191_0
Cartier, Pierre (1981), "La conjecture locale de Langlands pour GL(2) et la démonstration de Ph. Kutzko" (in French), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., 842, Berlin, New York: Springer-Verlag, pp. 112–138, doi:10.1007/BFb0089931, ISBN 978-3-540-10292-2, MR 636520
Gan, Wee Teck; Takeda, Shuichiro (2010), "The local Langlands conjecture for Sp(4)", International Mathematics Research Notices. IMRN 2010 (15): 2987–3038, doi:10.1093/imrn/rnp203, ISSN 1073-7928, MR arXiv:0805.2731 2673717[[arXiv]]:[[arXiv:0805.2731|0805.2731]]
Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) [1966], Representation theory and automorphic functions, Generalized functions, 6, Philadelphia, Pa.: W. B. Saunders Co., ISBN 978-0-12-279506-0, MR 0220673, http://books.google.com/books?id=_p5qAAAAMAAJ
Gan, Wee Teck; Takeda, Shuichiro (2011), arXiv:0706.0952v1, doi:10.4007/annals.2011.173.3.12
Harris, Michael; Taylor, Richard (2001), The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, ISBN 978-0-691-09090-0, MR 1876802, http://books.google.com/books?id=sigBbO69hvMC
Henniart, Guy (2000), "Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique", Inventiones Mathematicae 139 (2): 439–455, doi:10.1007/s002220050012, ISSN 0020-9910, MR 1738446
Henniart, Guy (2006), "On the local Langlands and Jacquet-Langlands correspondences", in Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis et al., International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 1171–1182, ISBN 978-3-03719-022-7, MR 2275640, http://mathunion.org/ICM/ICM2006.2/
Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, MR 0401654, http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book
Kudla, Stephen S. (1994), "The local Langlands correspondence: the non-Archimedean case", in Jannsen, Uwe; Kleiman, Steven; Serre, Jean-Pierre, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Providence, R.I.: American Mathematical Society, pp. 365–391, ISBN 978-0-8218-1637-0, MR 1265559
Kutzko, Philip (1980), "The Langlands conjecture for GL₂ of a local field", American Mathematical Society. Bulletin. New Series 2 (3): 455–458, doi:10.1090/S0273-0979-1980-14765-5, ISSN 0002-9904, MR 561532
Kutzko, Philip (1980), "The Langlands conjecture for Gl₂ of a local field", Annals of Mathematics. Second Series 112 (2): 381–412, doi:10.2307/1971151, ISSN 0003-486X, MR 592296
Langlands, Robert (1967), Letter to Prof. Weil, http://publications.ias.edu/rpl/section/21
Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614, http://publications.ias.edu/rpl/section/21
Langlands, Robert P. (1989) [1973], "On the classification of irreducible representations of real algebraic groups", in Sally, Paul J.; Vogan, David A., Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., 31, Providence, R.I.: American Mathematical Society, pp. 101–170, ISBN 978-0-8218-1526-7, MR 1011897, http://publications.ias.edu/rpl/paper/16
Laumon, G.; Rapoport, M.; Stuhler, U. (1993), "D-elliptic sheaves and the Langlands correspondence", Inventiones Mathematicae 113 (2): 217–338, doi:10.1007/BF01244308, ISSN 0020-9910, MR 1228127
Tunnell, Jerrold B. (1978), "On the local Langlands conjecture for GL(2)", Inventiones Mathematicae 46 (2): 179–200, doi:10.1007/BF01393255, ISSN 0020-9910, MR 0476703
Vogan, David A. (1993), "The local Langlands conjecture", in Adams, Jeffrey; Herb, Rebecca; Kudla, Stephen et al., Representation theory of groups and algebras, Contemp. Math., 145, Providence, R.I.: American Mathematical Society, pp. 305–379, ISBN 978-0-8218-5168-5, MR 1216197, http://www-math.mit.edu/~dav/paper.html
Wedhorn, Torsten (2008), "The local Langlands correspondence for GL(n) over p-adic fields", in Göttsche, Lothar; Harder, G.; Raghunathan, M. S., School on Automorphic Forms on GL(n), ICTP Lect. Notes, 21, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 237–320, arXiv:math/0011210, ISBN 9789295003378, MR 2508771, http://publications.ictp.it/lns/vol21/vol21toc.html
Weil, André (1974), "Exercices dyadiques", Inventiones Mathematicae 27: 1–22, doi:10.1007/BF01389962, ISSN 0020-9910, MR 0379445

External links

Harris, Michael (2000), The local Langlands correspondence, Notes of (half) a course at the IHP, http://www.math.jussieu.fr/~harris/IHPcourse.pdf
The work of Robert Langlands
Automorphic Forms - The local Langlands conjecture Lecture by Richard Taylor

Local Langlands conjectures

Contents

Local Langlands conjectures for GL1