nLab
local Langlands conjecture

Contents

Contents

Idea

The local Langlands conjectures are certain conjectures in the context of the Langlands program. Where the genuine Langlands correspondence concerns global fields, the local Langlands correspondence concerns local fields.

In the case of the algebraic group GL nGL_n (the general linear group), the local Langlands conjectures assert a correspondence between:

  1. F-semisimple Weil-Deligne representations (see Def. below) of the Weil group of a local field FF;

  2. irreducible admissible representations of GL n(F)GL_n(F) (see Def. below),

generalizing local class field theory from abelian Galois groups to non-abelian Galois groups.

More generally, it states that for a local field FF and a reductive group GG, the isomorphism classes of irreducible admissible representations of G(F)G(F) are partitioned into L-packets by the equivalence classes of L-parameters.

Definitions:

We discuss some of the basic definitions, see also Tate 77.

Throughout:

Definition

(irreducible and admissible representations) A linear representation of G(F)G(F) on a vector space VV is called:

Definition

(Weil group)
The Weil group W FW_F is the subgroup of the Galois group Gal(F¯/F)\mathrm{Gal}(\overline{F}/F) defined as the inverse image of Frobenius automorphisms Frob Gal(κ¯/κ)\mathrm{Frob}^{\mathbb{Z}}\subset \mathrm{Gal}(\overline{\kappa}/\kappa) under the surjective map Gal(F¯/F)Gal(κ¯/κ)\mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa).

Definition

(Weil-Deligne representation)
A Weil-Deligne representation is a pair (ρ 0,N)(\rho_{0},N) where ρ 0:W FGL n()\rho_{0}:W_{F}\to GL_{n}(\mathbb{C}) is a representation of the Weil group (Def. ) and NN is a nilpotent monodromy operator satisfying ρ 0(σ)Nρ 0(σ) 1=σN\rho_{0}(\sigma)N\rho_{0}(\sigma)^{-1}=\left\Vert\sigma\right\Vert N for all σW F\sigma\in W_{F}, where σ\Vert\sigma\Vert is the valuation of the corresponding element of F ×F^{\times} under the isomorphism of local class field theory.

Examples

Local Langlands correspondence for GL 1GL_1

Let FF be a pp-adic field. The only irreducible admissible representations of GL 1(F)GL_1(F) (=F ×=F^\times) are continuous group homomorphisms χ:F × ×\chi:F^\times\to\mathbb{C}^{\times}. The 11-dimensional Weil-Deligne representations (ρ 0,N)(\rho_{0},N) of W FW_F must have monodromy operator N=0N=0 and must factor through the abelianization W F abW_F^{\mathrm{ab}}. The local Langlands correspondence in this case is the same as local class field theory, and sends χ\chi to the Weil-Deligne representation (ρ 0,0)(\rho_{0},0), where ρ 0\rho_{0} is the composition W FW F abrec 1F ×χ ×W_{F}\to W_{F}^{\mathrm{ab}}\xrightarrow{\mathrm{rec}^{-1}}F^{\times}\xrightarrow{\chi}\mathbb{C}^{\times}, with rec:F ×W F ab\mathrm{rec}:F^{\times}\xrightarrow{\sim}W_{F}^{\mathrm{ab}} being the Artin reciprocity map of local class field theory.

Local Langlands correspondence for GL 2GL_2

Definition

Let FF be a pp-adic field, p2p\neq 2. Given continuous admissible characters χ 1\chi_{1} and χ 2\chi_{2}, we define I(χ 1,χ 2)I(\chi_{1},\chi_{2}) to be the vector space of function ϕ:GL 2(F)\phi:GL_{2}(F)\to \mathbb{C} which are locally constant with respect to the pp-adic topology on GL 2(F)GL_{2}(F) and satisfying

ϕ((a b 0 d)g)=χ 1(a)χ 2(d)a/d 1/2ϕ(g).\phi\left(\begin{pmatrix}a & b \\0 & d\end{pmatrix}g\right)=\chi_{1}(a)\chi_{2}(d)\Vert a/d\Vert^{1/2}\phi(g).

Letting GL 2(F)GL_2(F) act on this by translation gives us a representation of GL 2(F)GL_2(F). This representation is said to belong to the principal series.

Whether a principal series is irreducible or not is governed by the following theorem:

Theorem

[Bernstein-Zelevinsky]

If χ 1/χ 2 ±1\chi_{1}/\chi_{2}\neq\Vert\cdot\Vert^{\pm 1} then I(χ 1,χ 2)I(\chi_{1},\chi_{2}) is irreducible.

If χ 1/χ 2= 1\chi_{1}/\chi_{2}=\Vert\cdot\Vert^{1} then we have an exact sequence

0ρI(χ 1,χ 2)S(χ 1,χ 2)00\to\rho\to I(\chi_{1},\chi_{2})\to S(\chi_{1},\chi_{2})\to 0

where ρ\rho is the 11-dimensional representation of GL 2(F)GL_2(F) given by χ 1 1/2det\chi_{1}\Vert\cdot\Vert^{1/2}\det, and S(χ 1,χ 2)S(\chi_{1},\chi_{2}) is an irreducible representation.

If χ 1/χ 2= 1\chi_{1}/\chi_{2}=\Vert\cdot\Vert^{-1} then we have an exact sequence

0S(χ 1,χ 2)I(χ 1,χ 2)ρ00\to S(\chi_{1},\chi_{2})\to I(\chi_{1},\chi_{2})\to \rho\to 0

with ρ\rho and S(χ 1,χ 2)S(\chi_{1},\chi_{2}) as above.

Therefore the irreducible admissible representations of GL 2(F)GL_2(F) are the following:

  • Principal series representations I(χ 1,χ 2)I(\chi_{1},\chi_{2}) for χ 1/χ 2 ±1\chi_{1}/\chi_{2}\neq \Vert\cdot\Vert^{\pm 1}.

  • Special representations S(χ 1,χ 1×)S(\chi_{1},\chi_{1}\times\Vert\cdot\Vert).

  • 11-dimensional representations χdet\chi\circ\det.

  • “Base change” representations BC E F(ψ)BC_{E}^{F}(\psi) for EE a quadratic extension of FF and ψ\psi an admissible character ψ:E ×\psi:E\to\mathbb{C}^{\times}.

Let ρ i:W F ×\rho_{i}:W_{F}\to \mathbb{C}^{\times} be the representation of the Weil group associated to the character χ i:F × ×\chi_{i}:F^{\times}\to \mathbb{C}^{\times} by the local Langlands correspondence for GL 1GL_1. Then the local Langlands correspondence associates to each irreducible admissible representation of GL 2(F)GL_2(F) a 22-dimensional Weil-Deligne representation as follows:

  • To the principal series representation I(χ 1,χ 2)I(\chi_{1},\chi_{2}) we associate the Weil-Deligne representation (ρ 1ρ 2,0)(\rho_{1}\oplus\rho_{2},0).

  • To the special representation S(χ 1,χ 1×)S(\chi_{1},\chi_{1}\times\Vert\cdot\Vert), we associate the Weil-Deligne representation ((ρ 1 0 0 ρ 1),(0 1 0 0))\left(\begin{pmatrix}\Vert\cdot\Vert\rho_{1} & 0\\0 & \rho_{1}\end{pmatrix},\begin{pmatrix} 0 & 1\\0 & 0\end{pmatrix}\right).

  • To the 11-dimensional representation χ 1det\chi_{1}\circ\det, we associate the Weil-Deligne representation ((ρ 1× 1/2 0 0 ρ 1× 1/2),0)\left(\begin{pmatrix}\rho_{1}\times\Vert\cdot\Vert^{1/2} & 0\\0 & \rho_{1}\times\Vert\cdot\Vert^{-1/2}\end{pmatrix},0\right).

  • To the “base change” representation BC E F(ψ)BC_{E}^{F}(\psi) we associate the Weil-Deligne representation (Ind W E W Fσ,0Ind_{W_{E}}^{W_{F}}\sigma,0), where σ\sigma is the unique nontrivial element of Gal(E/F)\mathrm{Gal}(E/F).

Geometrization

Fargues-Scholze have developed a geometric approach to the local Langlands conjecture in FarguesScholze21. In this formulation the local Langlands correspondence is expressed as an equivalence between the category of certain \ell-adic sheaves on the moduli stack Bun G\mathrm{Bun}_{G} of GG-torsors on the Fargues-Fontaine curve on one hand, and the category of certain coherent sheaves on the moduli stack of L-parameters in the other (compare geometric Langlands correspondence). More precisely:

Conjecture

There is an equivalence

𝒟(Bun G,𝒪 L) ω𝒟 coh,Nilp b,qc(Z 1(W F,G^) 𝒪 L/G^)\mathcal{D}(\mathrm{Bun}_{G},\mathcal{O}_{L})^{\omega}\cong\mathcal{D}_{\mathrm{coh},\mathrm{Nilp}}^{b,\mathrm{qc}}(Z^{1}(W_{F},\widehat{G})_{\mathcal{O}_{L}}/\widehat{G})

of stable \infty-categories equipped with actions of Perf(Z 1(W F,G^) 𝒪 L/G^)\Perf(Z^{1}(W_{F},\widehat{G})_{\mathcal{O}_{L}}/\widehat{G}).

Fargues-Scholze are also able to associate an L-parameter φ A:W FG^( )\varphi_{A}:W_{F}\to\widehat{G}(\mathbb{Q}_{\ell}) to a Schur object AA in D(Bun G,¯ )D(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell}) using the method of V. Lafforgue involving excursion operators.

pp-adic and mod pp local Langlands

The pp-adic (resp. mod pp) local Langlands correspondence concerns pp-adic (resp. mod pp) representations of the absolute Galois group of a pp-adic field FF, as opposed to complex Weil-Deligne or \ell-adic representations. It is related to p-adic Hodge theory. Currently the only known case is the case GL 2( p)GL_2(\mathbb{Q}_{p}), see also Breuil2010.

References

See also:

Basic definitions are discussed in:

  • John Tate, Number theoretic background, in: Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI (ISBN:978-0-8218-3371-1, pdf, pdf)

An approach via the Fargues-Fontaine curve:

The pp-adic and mod pp local Langlands correspondence is discussed in:

  • Christophe Breuil, The emerging p-adic Langlands program, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010 (pdf)

Last revised on April 23, 2021 at 22:48:55. See the history of this page for a list of all contributions to it.