In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the local Langlands conjectures, introduced by , are part of 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 ...
. They describe a correspondence between the complex representations of a reductive
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
''G'' over a local field ''F'', and representations of the
Langlands group
In mathematics, the Langlands group is a conjectural group ''L'F'' attached to each local or global field ''F'', that satisfies properties similar to those of the Weil group. It was given that name by Robert Kottwitz. In Kottwitz's formulatio ...
of ''F'' into the L-group of ''G''. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of
local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...
from abelian
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
s to non-abelian Galois groups.
Local Langlands conjectures for GL1
The local Langlands conjectures for GL
1(''K'') follow from (and are essentially equivalent to)
local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...
. More precisely the
Artin map The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line ...
gives an isomorphism from the group GL
1(''K'')= ''K''
* to the abelianization of the
Weil group
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite leve ...
. In particular irreducible smooth representations of GL
1(''K'') are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL
1(C). This gives the Langlands correspondence between homomorphisms of the Weil group to GL
1(C) and
irreducible smooth representations of GL
1(''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 representation 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 complex representations are automatically admissible.
The
Bernstein–Zelevinsky classification In mathematics, the Bernstein–Zelevinsky classification, introduced by and , classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations.
References
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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).
described the irreducible admissible representations of general linear groups over local fields.
Local Langlands conjectures for GL2
The local Langlands conjecture for GL
2 of a local field says that there is a (unique) bijection π from 2-dimensional semisimple Weil-Deligne representations of the Weil group to irreducible smooth representations of GL
2(''F'') that preserves ''L''-functions, ε-factors, and commutes with twisting by characters of ''F''
*.
verified the local Langlands conjectures for GL
2 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. classified the smooth irreducible representations of GL
2(''F'') when ''F'' has odd residue characteristic (see also ), and claimed incorrectly that the classification for even residue characteristic differs only insignifictanly from the odd residue characteristic case.
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
2(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.)
proved the local Langlands conjectures for the general linear group GL
2(''K'') over the 2-adic numbers, and over local fields containing a cube root of unity.
proved the local Langlands conjectures for the general linear group GL
2(''K'') over all local fields.
and 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)
proved the local Langlands conjectures for the general linear group GL
''n''(''K'') for positive characteristic local fields ''K''. gave an exposition of their work.
proved the local Langlands conjectures for the general linear group GL
''n''(''K'') for characteristic 0 local fields ''K''. gave another proof. and gave expositions of their work.
Local Langlands conjectures for other groups
and discuss the Langlands conjectures for more general groups. The Langlands conjectures for arbitrary reductive groups ''G'' are more complicated to state than the ones for general linear groups, and it is unclear what the best 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
local Langlands group to the
''L''-group of ''G''. Some earlier versions used the Weil−Deligne group or the Weil group instead of the local Langlands group, which gives a slightly weaker form of the conjecture.
proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the
Langlands classification In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group ''G'', suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One ...
of their irreducible admissible representations (up to infinitesimal equivalence), or, equivalently, of their irreducible
-modules.
proved the local Langlands conjectures for the
symplectic similitude group GSp(4) and used that in to deduce it for the
symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic grou ...
Sp(4).
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External links
*{{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
The work of Robert Langlands Automorphic Forms - The local Langlands conjectureLecture by Richard Taylor
Zeta and L-functions
Representation theory of Lie groups
Automorphic forms
Conjectures
Class field theory
Langlands program