Local Langlands for $\mathrm{GL}_n$¶

Job: match irreducible smooth representations of $\mathrm{GL}_n(F)$ ($F$ local) with $n$-dimensional Frobenius-semisimple Weil–Deligne representations — the local dictionary that constrains global reciprocity.

Core¶

Setup¶

Let $F$ be a nonarchimedean local field (finite extension of $\mathbb{Q}_p$ or $\mathbb{F}_q((t))$), residue field $\mathbb{F}_q$, Weil group $W_F\subset\mathrm{Gal}(\overline{F}/F)$. Smooth representations of $G=\mathrm{GL}_n(F)$: every vector fixed by some open compact subgroup; irreducible admissible $\Leftrightarrow$ irreducible smooth of finite length Jacquet modules (automatic for $\mathrm{GL}_n$ in the irreducible case under standard hypotheses).

Archimedean local Langlands uses $(\mathfrak{g},K)$-modules / real reductive groups (Langlands classification via tempered parameters); stated separately below in brief.

Weil–Deligne representations¶

A Weil–Deligne representation of $W_F$ on a finite-dimensional $\mathbb{C}$-vector space $V$ is a pair $(\rho,N)$ where $\rho:W_F\to\mathrm{GL}(V)$ is smooth (open kernel on inertia after restriction) and continuous on $W_F/I$ with $I$ inertia, and $N\in\operatorname{End}(V)$ is nilpotent with $$ \rho(w)N\rho(w)^{-1}=|w|\,N,\qquad |w|=q^{-v(w)}. $$ Frobenius-semisimple: $\rho(\Phi)$ semisimple for geometric Frobenius $\Phi$. Complex dual of $\ell$-adic Galois representations (via Grothendieck’s monodromy theorem) lands in WD-representations: $N$ encodes monodromy, $\rho$ the semisimple Weil action.

Feature	Content
Local $L$- / $\varepsilon$-factor	From WD data $(\rho,N)$ via Deligne / Langlands–Deligne recipes
Unramified $\rho$	$N=0$, $\rho$ factors through $W_F/I\cong\mathbb{Z}$
Special / Steinberg type	nontrivial $N$ on indecomposable Jordan blocks

Local Langlands correspondence (LLC) for $\mathrm{GL}_n$¶

Theorem (Harris–Taylor, Henniart; Scholze simplification). There is a unique bijection $$ \operatorname{Irr}\bigl(\mathrm{GL}_n(F)\bigr)\;\longleftrightarrow\;\bigl{\text{$n$-dim.\ Frob-ss WD representations of $W_F$}\bigr}/\cong $$ characterized by matching of $L$- and $\varepsilon$-factors of pairs (Rankin–Selberg / Godement–Jacquet on the automorphic side; Artin–Deligne on the Galois side), and compatible with twists by characters, central characters $\leftrightarrow\det$, and $n=1$ (local class field theory).

Properties used as axioms (Henniart uniqueness):

$\pi\leftrightarrow\phi_\pi$ implies $\pi\otimes(\chi\circ\det)\leftrightarrow\phi_\pi\otimes\chi$ for characters $\chi$ of $F^\times\cong W_F^{\mathrm{ab}}$.
$\omega_\pi=\det\phi_\pi$ under local CFT.
$L(s,\pi\times\pi')=L(s,\phi_\pi\otimes\phi_{\pi'})$ and $\varepsilon(s,\pi\times\pi',\psi)=\varepsilon(s,\phi_\pi\otimes\phi_{\pi'},\psi)$ for all irreducible $\pi'$.

Unramified representations and Satake¶

Let $K=\mathrm{GL}_n(\mathcal{O}_F)$. An irreducible smooth $\pi$ is unramified (spherical) if $\pi^K\neq 0$. Then $\dim\pi^K=1$ and the Satake isomorphism identifies the Hecke algebra $$ \mathcal{H}(G,K)=C_c^\infty(K\backslash G/K)\xrightarrow{\sim}\mathbb{C}[x_1^{\pm1},\ldots,x_n^{\pm1}]^{S_n}. $$ Thus unramified $\pi$ $\leftrightarrow$ unordered $n$-tuples $(\alpha_1,\ldots,\alpha_n)\in(\mathbb{C}^\times)^n/S_n$ (Satake parameters), equivalently semisimple conjugacy classes in $\mathrm{GL}_n(\mathbb{C})=\widehat{\mathrm{GL}_n}$.

LLC unramified: $\phi_\pi$ is unramified ($N=0$, inertia acts trivially) with $\phi_\pi(\mathrm{Frob})$ of eigenvalues $\alpha_i$. Local $L$-factor: $$ L(s,\pi)=\prod_{i=1}^n\bigl(1-\alpha_i q^{-s}\bigr)^{-1}. $$ This is the local unramified Euler factor appearing in global $L(s,\pi)$.

Bernstein–Zelevinsky and classification sketch¶

Irreducible smooth reps of $\mathrm{GL}_n(F)$ are classified by multisegments (Zelevinsky) / Langlands quotients of normalized parabolic inductions of essentially square-integrable reps. Square-integrable reps correspond (under LLC) to irreducible WD-representations up to twist; segments $\leftrightarrow$ indecomposable WD blocks with monodromy. The Steinberg representation $\mathrm{St}_n$ matches the special WD-representation with irreducible restriction to $W_F$ and maximal monodromy $N$.

Archimedean sketch¶

For $F=\mathbb{R}$ or $\mathbb{C}$, Langlands parameters are continuous homomorphisms $\phi:W_F\to\mathrm{GL}_n(\mathbb{C})$ ($W_\mathbb{R}=\mathbb{C}^\times\rtimes\mathrm{Gal}(\mathbb{C}/\mathbb{R})$) up to conjugacy, matching irreducible Casselman–Wallach representations of $\mathrm{GL}_n(F)$. Infinity types of algebraic automorphic representations correspond to Hodge–Tate / cohomological parameters.

Beyond $\mathrm{GL}_n$ (pointer)¶

For general reductive $G$, LLC predicts packets of representations labeled by $L$-parameters $\phi:W_F'\to{}^LG$ (and enhancements for $A$-parameters). Proven in many cases (classical groups via Arthur, unitary groups, etc.) but packet structure and character identities replace bijections.

See it¶

Local class field theory as $n=1$. Irreducible smooth reps of $\mathrm{GL}_1(F)=F^\times$ are smooth characters $\chi:F^\times\to\mathbb{C}^\times$. LLC + local CFT: $\chi\leftrightarrow\phi_\chi:W_F\to\mathbb{C}^\times$ via $W_F^{\mathrm{ab}}\cong F^\times$. $L$-factors: if $\chi$ unramified, $\chi(\varpi)=\alpha$, then $L(s,\chi)=(1-\alpha q^{-s})^{-1}$; ramified $\Rightarrow L=1$. This is the seed of the entire local theory.

Conductors. Swan conductor of $\phi_\pi$ equals the Artin conductor of $\pi$ (Godement–Jacquet conductor); level-lowering and modularity lifting depend on this equality through residual representations and deformation rings.

Check¶

Why does matching $L$- and $\varepsilon$-factors of pairs $\pi\times\pi'$ determine LLC uniquely for $\mathrm{GL}_n$?

Henniart: the collection of twisted factors separates irreducible representations — roughly, Rankin–Selberg data as $\pi'$ varies determine the WD-parameter as a semisimple tensor class, using $n=1$ twists and induction on $n$. Uniqueness of a bijection satisfying the listed axioms follows; existence is the hard geometric input (Shimura varieties / perfectoid spaces).

What is the Satake parameter of the unramified principal series of $\mathrm{GL}_2(F)$ induced from unramified characters $(\chi_1,\chi_2)$?

Normalized parabolic induction $i_B(\chi_1\otimes\chi_2)$ (when irreducible) has Satake parameters $(\chi_1(\varpi),\chi_2(\varpi))$; $L(s,\pi)=(1-\chi_1(\varpi)q^{-s})^{-1}(1-\chi_2(\varpi)q^{-s})^{-1}$.

Pitfalls¶

Normalized vs non-normalized induction: LLC uses unitary/normalized conventions; off-by-$|\cdot|^{\pm 1/2}$ errors break $\varepsilon$-factor matching.
Frobenius-semisimple is mandatory: raw $\ell$-adic inertia action need not be semisimple; monodromy $N$ isolates the unipotent part.
Complex coefficients: LLC is usually stated over $\mathbb{C}$; comparison with $\overline{\mathbb{Q}}_\ell$-coefficients uses embeddings and purity.
“The” local component of a global $\pi$: restricted tensor product depends on choices of hyperspecial lattices at unramified places — standard but easy to muddy.

Refs¶

Harris–Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties
Henniart, Une preuve simple des conjectures de Langlands pour $\mathrm{GL}_n$ sur un corps $p$-adique
Scholze, The local Langlands correspondence for $\mathrm{GL}_n$ over $p$-adic fields
Kudla, The local Langlands correspondence: the non-Archimedean case (in Motives)
Bushnell–Henniart, The Local Langlands Conjecture for $\mathrm{GL}(2)$

Next¶

Langlands index · Global foundations · Class field · Representation theory · L-functions · Algebraic NT · Geometric Langlands (ramified) — local geometric Langlands / Fargues–Scholze

Feature	Content
Local \(L\)- / \(\varepsilon\)-factor	From WD data \((\rho,N)\) via Deligne / Langlands–Deligne recipes
Unramified \(\rho\)	\(N=0\), \(\rho\) factors through \(W_F/I\cong\mathbb{Z}\)
Special / Steinberg type	nontrivial \(N\) on indecomposable Jordan blocks

Local Langlands for \(\mathrm{GL}_n\)¶