Local Langlands conjectures

In mathematics, the local Langlands conjectures, introduced by Robert Langlands,¹² are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group ${\displaystyle G}$ over a local field ${\displaystyle F}$, and representations of the Langlands group of ${\displaystyle F}$ into the ${\displaystyle L}$-group of ${\displaystyle G}$. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.

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

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 ${\displaystyle V}$ together with a nilpotent endomorphism ${\displaystyle N}$ of ${\displaystyle V}$ such that ${\displaystyle wNw^{-1}=\|w\|N}$, or equivalently a representation of the Weil–Deligne group. In addition, the representation of the Weil group should have an open kernel and be (Frobenius) semisimple.

For every Frobenius semisimple complex ${\displaystyle n}$-dimensional Weil–Deligne representation ${\displaystyle \rho }$ of the Weil group of ${\displaystyle F}$ there is an L-function ${\displaystyle L(s,\rho )}$ and a local ε-factor ${\displaystyle \varepsilon (s,\rho ,\psi )}$ (depending on a character ${\displaystyle \psi }$ of ${\displaystyle F}$).

The representations of ${\displaystyle \operatorname {GL} _{n}(F)}$ 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 ${\displaystyle \pi }$ there is an L-function ${\displaystyle L(s,\pi )}$ and a local ${\displaystyle \varepsilon }$-factor ${\displaystyle \varepsilon (s,\pi ,\psi )}$ (depending on a character ${\displaystyle \psi }$ of ${\displaystyle F}$). More generally, if there are two irreducible admissible representations ${\displaystyle \pi }$ and ${\displaystyle \pi '}$ of general linear groups, there are local Rankin–Selberg convolution L-functions ${\displaystyle L(s,\pi \times \pi ')}$ and ${\displaystyle \varepsilon }$-factors ${\displaystyle \varepsilon (s,\pi \times \pi ',\psi )}$.

Bushnell and Kutzko³ described the irreducible admissible representations of general linear groups over local fields.

The local Langlands conjecture for ${\displaystyle \operatorname {GL} _{2}}$ of a local field says that there is a (unique) bijection ${\displaystyle \pi }$ from 2-dimensional semisimple Weil-Deligne representations of the Weil group to irreducible smooth representations of ${\displaystyle \operatorname {GL} _{2}(F)}$ that preserves L-functions, ${\displaystyle \varepsilon }$-factors, and commutes with twisting by characters of ${\displaystyle F^{\times }}$.

Jacquet and Langlands⁴ verified the local Langlands conjectures for ${\displaystyle \operatorname {GL} _{2}}$ 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.

Gelfand & Graev (1962) classified the smooth irreducible representations of ${\displaystyle \operatorname {GL} _{2}(F)}$ when ${\displaystyle F}$ 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. Weil⁵ 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 ${\displaystyle \operatorname {PGL} _{2}(\mathbb {C} )}$ 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.)

Tunnell⁶ proved the local Langlands conjectures for the general linear group ${\displaystyle \operatorname {GL} _{2}(K)}$ over the 2-adic numbers, and over local fields containing a cube root of unity. Kutzko⁷ proved the local Langlands conjectures for the general linear group ${\displaystyle \operatorname {GL} _{2}(K)}$ over all local fields.

Cartier,⁸ as well as Bushnell and Henniart,⁹ gave expositions of the proof.

The local Langlands conjectures for general linear groups state that there are unique bijections ${\displaystyle \pi \leftrightarrow \rho _{\pi }}$ from equivalence classes of irreducible admissible representations ${\displaystyle \pi }$ of ${\displaystyle \operatorname {GL} _{n}(F)}$ to equivalence classes of continuous Frobenius semisimple complex ${\displaystyle n}$-dimensional Weil–Deligne representations ${\displaystyle \rho _{\pi }}$ of the Weil group of ${\displaystyle F}$ that preserve L-functions and ${\displaystyle \varepsilon }$-factors of pairs of representations, and coincide with the Artin map for 1-dimensional representations. In other words,

• ${\displaystyle L(s,\rho _{\pi }\otimes \rho _{\pi '})=L(s,\pi \times \pi ')}$
• ${\displaystyle \varepsilon (s,\rho _{\pi }\otimes \rho _{\pi '},\psi )=\varepsilon (s,\pi \times \pi ',\psi ).}$

Laumon, Rapoport and Stuhler¹⁰ proved the local Langlands conjectures for the general linear group ${\displaystyle \operatorname {GL} _{n}(K)}$ for positive characteristic local fields ${\displaystyle K}$. Carayol¹¹ gave an exposition of their work.

Harris and Taylor¹² proved the local Langlands conjectures for the general linear group ${\displaystyle \operatorname {GL} _{n}(K)}$ for characteristic 0 local fields ${\displaystyle K}$. Henniart¹³ gave another proof. Carayol¹⁴ and Wedhorn¹⁵ gave expositions of their work.

Borel¹⁶ and Vogan¹⁷ discuss the Langlands conjectures for more general groups. The Langlands conjectures for arbitrary reductive groups ${\displaystyle G}$ 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 ${\displaystyle G}$. 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.

Langlands¹⁸ proved the Langlands conjectures for groups over ${\displaystyle \mathbb {R} }$ and ${\displaystyle \mathbb {C} }$ by giving the Langlands classification of their irreducible admissible representations (up to infinitesimal equivalence), or, equivalently, of their irreducible ${\displaystyle ({\mathfrak {g}},K)}$-modules.

Gan and Takeda¹⁹ proved the local Langlands conjectures for the symplectic similitude group ${\displaystyle \operatorname {GSp} (4)}$ and used that to deduce it for the symplectic group ${\displaystyle \operatorname {Sp} (4)}$.²⁰