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| In mathematics, the '''local Langlands conjectures''', introduced by {{harvs|txt|last=Langlands|year1=1967|year2=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 group]]s over the local field, generalizing [[local class field theory]] from abelian [[Galois group]]s to non-abelian Galois groups.
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| ==Local Langlands conjectures for GL<sub>1</sub>==
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| The local Langlands conjectures for GL<sub>1</sub>(''K'') follow from (and are essentially equivalent to) [[local class field theory]]. More precisely the [[Artin map]] gives an isomorphism from the group GL<sub>1</sub>(''K'')= ''K''<sup>*</sup> to the abelianization of the Weil group. In particular irreducible smooth representations of GL<sub>1</sub>(''K'') are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL<sub>1</sub>('''C'''). This gives the Langlands correspondence between homomorphisms of the Weil group to GL<sub>1</sub>('''C''') and | |
| irreducible smooth representations of GL<sub>1</sub>(''K'').
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| ==Representations of the Weil group==
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| 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''<sup>−1</sup>=||''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.
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| 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 [[Langlands–Deligne local constant|local ε-factor]] ε(''s'',ρ,ψ) (depending on a character ψ of ''F'').
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| ==Representations of GL<sub>''n''</sub>(''F'')==
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| The representations of GL<sub>''n''</sub>(''F'') appearing in the local Langlands correspondence are smooth irreducible complex representations.
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| *"Smooth" means that every vector is fixed by some open subgroup.
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| *"Irreducible" means that the representation is nonzero and has no subrepresentations other than 0 and itself.
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| Smooth irreducible representations are automatically admissible.
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| The [[Bernstein–Zelevinsky classification]] reduces the classification of irreducible smooth representations to cuspidal representations.
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| 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).
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| {{harvtxt|Bushnell|Kutzko|1993}} described the irreducible admissible representations of general linear groups over local fields.
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| ==Local Langlands conjectures for GL<sub>2</sub>==
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| The local Langlands conjecture for GL<sub>2</sub> 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 GL<sub>2</sub>(''F'') that preserves ''L''-functions, ε-factors, and commutes with twisting by characters of ''F''<sup>*</sup>.
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| {{harvtxt|Jacquet|Langlands|1970}} verified the local Langlands conjectures for GL<sub>2</sub> 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. {{harvtxt|Gelfand|Graev|1962}} classified the smooth irreducible representations of GL<sub>2</sub>(''F'') when ''F'' has odd residue characteristic (see also {{harv|Gelfand|Graev|Pyatetskii-Shapiro|1969|loc=chapter 2}}), and claimed incorrectly that the classification for even residue characteristic differs only insignifictanly from the odd residue characteristic case.
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| {{harvtxt|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 PGL<sub>2</sub>('''C''') is of tetrahedral or octahedral type. (For global Langlands conjectures, 2-dimensional representations can also be of icosahedral type, but this cannot happen in the local case as the Galois groups are solvable.)
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| {{harvtxt|Tunnell|1978}} proved the local Langlands conjectures for the general linear group GL<sub>2</sub>(''K'') over the 2-adic numbers, and over local fields containing a cube root of unity.
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| {{harvs|txt|last=Kutzko|year1=1980|year2=1980b}} proved the local Langlands conjectures for the general linear group GL<sub>2</sub>(''K'') over all local fields.
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| {{harvtxt|Cartier|1981}} and {{harvtxt|Bushnell|Henniart|2006}} gave expositions of the proof.
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| ==Local Langlands conjectures for GL<sub>n</sub>==
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| The local Langlands conjectures for general linear groups state that there are unique bijections π ↔ ρ<sub>π</sub> from equivalence classes of irreducible admissible representations π of GL<sub>''n''</sub>(''F'') to equivalence classes of continuous Frobenius semisimple complex ''n''-dimensional Weil–Deligne representations ρ<sub>π</sub> 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,
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| *L(''s'',ρ<sub>π</sub>⊗ρ<sub>π'</sub>) = L(''s'',π×π')
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| *ε(''s'',ρ<sub>π</sub>⊗ρ<sub>π'</sub>,ψ) = ε(''s'',π×π',ψ)
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| {{harvtxt|Laumon|Rapoport|Stuhler|1993}} proved the local Langlands conjectures for the general linear group GL<sub>''n''</sub>(''K'') for positive characteristic local fields ''K''. {{harvtxt|Carayol|1992}} gave an exposition of their work.
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| {{harvs|txt|authorlink=Richard Taylor (mathematician)|first=Richard|last= Taylor|first2= Michael|last2= Harris|year=2001}} proved the local Langlands conjectures for the general linear group GL<sub>''n''</sub>(''K'') for characteristic 0 local fields ''K''. {{harvtxt|Henniart|2001}} gave another proof. {{harvtxt|Carayol|2000}} and {{harvtxt|Wedhorn|2008}} gave expositions of their work.
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| ==Local Langlands conjectures for other groups==
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| {{harvtxt|Borel|1979}} and {{harvtxt|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 [[Langlands dual|''L''-group]] of ''G''.
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| {{harvtxt|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 [[(g,K)-module|<math>(\mathfrak{g},K)</math>-modules]].
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| {{harvtxt|Gan|Takeda|2011}} proved the local Langlands conjectures for the [[symplectic similitude group]] GSp(4) and used that in {{harvtxt|Gan|Takeda|2010}} to deduce it for the [[symplectic group]] Sp(4).
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| ==References==
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| *{{Citation | last1=Kutzko | first1=Philip | title=The Langlands conjecture for GL<sub>2</sub> of a local field | doi=10.1090/S0273-0979-1980-14765-5 | mr=561532 | year=1980 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=2 | issue=3 | pages=455–458}}
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| *{{Citation | last1=Kutzko | first1=Philip | title=The Langlands conjecture for Gl<sub>2</sub> of a local field | doi=10.2307/1971151 | mr=592296 | year=1980 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=112 | issue=2 | pages=381–412}}
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| *{{Citation | last1=Vogan | first1=David A. | editor1-last=Adams | editor1-first=Jeffrey | editor2-last=Herb | editor2-first=Rebecca | editor3-last=Kudla | editor3-first=Stephen | editor4-last=Li | editor4-first=Jian-Shu | editor5-last=Lipsman | editor5-first=Ron | editor6-last=Rosenberg | editor6-first=Jonathan | title=Representation theory of groups and algebras | url=http://www-math.mit.edu/~dav/paper.html | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Contemp. Math. | isbn=978-0-8218-5168-5 | mr=1216197 | year=1993 | volume=145 | chapter=The local Langlands conjecture | pages=305–379}}
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| *{{Citation | last1=Wedhorn | first1=Torsten | editor1-last=Göttsche | editor1-first=Lothar | editor2-last=Harder | editor2-first=G. | editor3-last=Raghunathan | editor3-first=M. S. | title=School on Automorphic Forms on GL(n) | url=http://publications.ictp.it/lns/vol21/vol21toc.html | publisher=Abdus Salam Int. Cent. Theoret. Phys., Trieste | series=ICTP Lect. Notes | isbn=978-92-95003-37-8 | mr=2508771 |arxiv=math/0011210 | year=2008 | volume=21 | chapter=The local Langlands correspondence for GL(n) over p-adic fields | pages=237–320}}
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| *{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Exercices dyadiques | doi=10.1007/BF01389962 | mr=0379445 | year=1974 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=27 | pages=1–22}}
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| ==External links==
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| *{{citation|first=Michael|last=Harris|url=http://www.math.jussieu.fr/~harris/IHPcourse.pdf|year=2000|title=The local Langlands correspondence|series=Notes of (half) a course at the IHP}}
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| *[http://publications.ias.edu/rpl/ The work of Robert Langlands ]
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| *[http://video.ias.edu/Automorphic-Forms-taylor Automorphic Forms - The local Langlands conjecture] Lecture by Richard Taylor
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| [[Category:Zeta and L-functions]]
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| [[Category:Number theory]]
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| [[Category:Representation theory of Lie groups]]
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| [[Category:Automorphic forms]]
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| [[Category:Conjectures]]
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| [[Category:Class field theory]]
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| [[Category:Langlands program]]
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