Local Langlands correspondence for $\mathrm{GL}_n$

Statement

Let $F$ be a nonarchimedean local field. There is a unique canonical bijection

\operatorname{rec}_{F,n}: \operatorname{Irr}\bigl(\operatorname{GL}_n(F)\bigr) \xrightarrow{\ \sim\ } \operatorname{WD}_n(F)

from irreducible smooth complex representations of $\operatorname{GL}_n(F)$ to isomorphism classes of $n$ -dimensional Frobenius-semisimple Weil–Deligne representations, normalized so that $n=1$ agrees with local class field theory and so that standard local $L$ - and epsilon factors agree.

Packet consequence

Every $L$ -packet for $\operatorname{GL}_n$ is a singleton. Thus the basic local correspondence is an actual bijection rather than a finite-to-one map.

Compatibility properties

The correspondence is compatible with twisting by characters and taking contragredients. The determinant of the parameter corresponds, through local class field theory, to the central character of the representation. Under the p-adic Langlands classification, direct sums of Weil–Deligne parameters correspond to the appropriate irreducible quotients of normalized parabolic inductions.

Special classes

Supercuspidal representations correspond to irreducible $n$ -dimensional representations of $W_F$ , necessarily with zero monodromy. Essentially Essentially square-integrable representations correspond to indecomposable Weil–Deligne representations. Unramified representations correspond to parameters trivial on the inertia subgroup with zero monodromy.

Proof history

The theorem was proved through the work of Laumon–Rapoport–Stuhler and Harris–Taylor, with Henniart establishing the numerical correspondence and a characterization by local factors. Different constructions are known to agree under the standard normalization.

References

Michael Harris and Richard Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Princeton University Press, 2001. Publisher.
Guy Henniart, “Une preuve simple des conjectures de Langlands pour $\mathrm{GL}(n)$ sur un corps $p$ -adique,” Inventiones Mathematicae 139 (2000), 439–455. DOI.
Michael Harris, “On the local Langlands correspondence,” 2003. arXiv.