Local Langlands conjectures

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 {{nowrap|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 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 Langlands–Deligne local constant ε(s,ρ,ψ) (depending on a character ψ of 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 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 {{nowrap|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 {{nowrap|ε-factors}} ε(s,π×π',ψ).

{{harvtxt|Bushnell|Kutzko|1993}} described the irreducible admissible representations of general linear groups over local fields.

The local Langlands conjecture for GL₂ 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 GL₂(F) that preserves L-functions, {{nowrap|ε-factors}}, and commutes with twisting by characters of F^*.

{{harvtxt|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. {{harvtxt|Gelfand|Graev|1962}} classified the smooth irreducible representations of GL₂(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.

{{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 PGL₂(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.)

{{harvtxt|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.

{{harvs|txt|last=Kutzko|year1=1980|year2=1980b}} proved the local Langlands conjectures for the general linear group GL₂(K) over all local fields.

{{harvtxt|Cartier|1981}} and {{harvtxt|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 {{nowrap|ε-factors}} of pairs of representations, and coincide with the Artin map for {{nowrap|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)

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

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

{{harvtxt|Borel|1979}} and {{harvtxt|Vogan|1993}} 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.

{{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 $(\backslash mathfrak\{g\},K)$-modules.

{{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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• {{citation|first=Michael|last=Harris|url=https://webusers.imj-prg.fr/~michael.harris/IHPcourse.pdf|year=2000|title=The local Langlands correspondence|series=Notes of (half) a course at the IHP}}
• [http://publications.ias.edu/rpl/ The work of Robert Langlands ]
• [http://video.ias.edu/Automorphic-Forms-taylor Automorphic Forms - The local Langlands conjecture] Lecture by Richard Taylor

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