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 $GL_n$ (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 $F$;

2. irreducible admissible representations of $GL_n(F)$ (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 $F$ and a reductive group $G$, the isomorphism classes of irreducible admissible representations of $G(F)$ 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:

• $F$ is a p-adic field with $\kappa$ denoting its residue field;

• $G$ is a reductive group.

###### Definition

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

• irreducible if the only $G(F)$-invariant subspaces are $0$ and $V$.

• admissible if for any open subgroup $U$ of $G(F)$ the fixed subspace $V^{U}$ is finite-dimensional.

###### Definition

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

###### Definition

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

## Examples

### Local Langlands correspondence for $GL_1$

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

### Local Langlands correspondence for $GL_2$

###### Definition

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

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

Letting $GL_2(F)$ act on this by translation gives us a representation of $GL_2(F)$. 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 $\chi_{1}/\chi_{2}\neq\Vert\cdot\Vert^{\pm 1}$ then $I(\chi_{1},\chi_{2})$ is irreducible.

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

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

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

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

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

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

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

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

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

• $1$-dimensional representations $\chi\circ\det$.

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

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

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

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

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

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

## Geometrization

Laurent Fargues and Peter 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 derived category of certain $\ell$-adic sheaves (this requires the theory of solid abelian groups? from condensed mathematics) on the moduli stack $\mathrm{Bun}_{G}$ of $G$-torsors on the Fargues-Fontaine curve on one hand, and the derived category of certain coherent sheaves on the moduli stack of L-parameters (denoted $Z^{1}(W_{F},\widehat{G})_{\mathcal{O}_{L}}/\widehat{G}$ in FarguesScholze21) in the other (compare geometric Langlands correspondence). More precisely:

###### Conjecture

There is an equivalence

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

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

In their work Fargues and Scholze are able to associate an L-parameter $\varphi_{A}:W_{F}\to\widehat{G}(\mathbb{Q}_{\ell})$ to a Schur object $A$ in $D(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ using the method of Vincent Lafforgue (originally developed for the global Langlands correspondence for function fields over $\mathbb{F}_{q}$) involving excursion operators. This is also known as the “automorphic to Galois” direction in the global setting.

In the other direction (analogous to the “Galois to automorphic direction”) Fargues and Scholze approach this via the construction of the spectral action, which involves the use of infinity groupoids? (which they refer to as anima). Subsequent work of Johannes Anschütz and Arthur-Cesar Le Bras in AnschutzLeBras21 use this spectral action to show the existence of the Hecke eigensheaf.

## $p$-adic and mod $p$ local Langlands

The $p$-adic (resp. mod $p$) local Langlands correspondence concerns $p$-adic (resp. mod $p$) representations of the absolute Galois group of a $p$-adic field $F$, 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 $GL_2(\mathbb{Q}_{p})$, see also Breuil2010.

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 above work is partially inspired by the following work on the global Langlands correspondence for function fields over $\mathbb{F}_{q}$:

The spectral action of Fargues and Scholze is used to show the existence of the Hecke eigensheaf in

• Johannes Anschütz and Arthur-Cesar Le Bras, Averaging functors in Fargues’ program for GLn_ arXiv:2104.04701

The $p$-adic and mod $p$ 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)