Fargues-Scholze parameter map

Let $E$ be a nonarchimedean local field of residue characteristic $p$ , let $G$ be a connected reductive $E$ -group, and let $\ell\neq p$ . After the coefficient and square-root-of- $q$ normalizations in Fargues–Scholze theory, every irreducible smooth $\overline{\mathbb Q}_\ell$ -representation $\pi$ of $G(E)$ has a canonically attached conjugacy class of continuous semisimple parameter

\varphi_\pi^{\mathrm{FS}}: W_E\longrightarrow{}^LG(\overline{\mathbb Q}_\ell).

Here $W_E$ is the Weil group and ${}^LG$ is the $L$ -group. This is the Fargues–Scholze parameter map.

Construction

The spectral Bernstein center acts on the ordinary Bernstein center through local excursion operators. Schur's lemma evaluates those central operators on an irreducible representation. The resulting system of invariant scalars reconstructs a unique semisimple $L$ -parameter.

Proven compatibilities

The map agrees with local class field theory for tori and, for $\operatorname{GL}_n$ , with the semisimplification of the established local Langlands correspondence. It is compatible with products, central characters, twists, contragredients, restriction along maps inducing an isomorphism on adjoint groups, Weil restriction, and normalized parabolic induction in the precise formulations of the theorem.

Exact limitation

The construction is not an unconditional full refined local Langlands correspondence for every $G$ . The parameter is semisimple: it does not in general recover the monodromy operator or the Deligne $\operatorname{SL}_2$ , and it does not prove surjectivity, finite packets, packet enhancements, endoscopic character identities, or all expected local factors in full generality.

For $\operatorname{GL}_n$ , compatibility with parabolic induction and the supercuspidal correspondence identifies it with the usual semisimple parameter for every irreducible smooth representation.

References

Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” Theorem I.9.6 and Chapter IX. arXiv.
Naoki Imai, “On the geometrization of the local Langlands correspondence,” 2024. arXiv.