Vincent Lafforgue global parameterization

Let $F=\mathbb F_q(X)$ be the function field of a smooth projective geometrically connected curve, let $G$ be a connected reductive $F$ -group, and fix level and central data. The Vincent Lafforgue global parameterization is a canonical decomposition of the finite-dimensional space of cuspidal automorphic forms

\mathcal A_{\mathrm{cusp}} = \bigoplus_\sigma \mathcal H_\sigma

indexed by conjugacy classes of continuous semisimple global $\ell$ -adic Langlands parameters $\sigma$ , unramified away from the level.

For split $G$ ,

\sigma: \operatorname{Gal}(\overline F/F) \longrightarrow \widehat G(\overline{\mathbb Q}_\ell).

The source is the absolute Galois group of $F$ , and $\widehat G$ is the Langlands dual group.

For nonsplit $G$ , the target and projection condition use the $L$ -group; the automorphic side naturally includes the relevant inner forms.

Characterizing property

For every unramified place $v$ and every algebraic representation $V$ of $\widehat G$ , the spherical Hecke operator attached to $V$ acts on $\mathcal H_\sigma$ through the scalar

\operatorname{tr}\!\left(V(\sigma(\operatorname{Frob}_v))\right),

with Frobenius and Satake normalizations chosen consistently.

Construction

Cohomology of multiple-leg $G$ -shtukas, geometric Satake, partial Frobenius, and coalescence produce the commuting excursion algebra. Its generalized characters reconstruct the semisimple parameters and define the displayed summands.

Exact scope

This is the automorphic-to-Galois direction for arbitrary reductive groups over global function fields. It is stronger than merely attaching almost-all Satake classes, but it is not a general bijection between individual automorphic representations and parameters and does not provide the missing Arthur multiplicity formulas. For $\operatorname{GL}_n$ , Drinfeld and Laurent Lafforgue prove the fuller correspondence.

The decomposition is $\ell$ -adic; independence of $\ell$ in the broad general setting is an additional conjectural issue.

References

Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” JAMS 31 (2018), 719–891. arXiv.
Vincent Lafforgue, “Introduction to chtoucas for reductive groups and to the global Langlands parameterization.” arXiv.