The plectic conjecture over function fields

TL;DR

This paper proves the plectic conjecture over global function fields, establishing new compatibilities between Weil group actions and Hecke actions on moduli spaces of shtukas, extending prior structures in the Langlands program.

Contribution

It proves the plectic conjecture over function fields, extending the Weil group action structure on the cohomology of shtukas and demonstrating its compatibility with Hecke operators.

Findings

Weil group actions commute with Hecke actions on shtuka cohomology.

Provides a moduli-theoretic description of Frobenius actions.

Extends the structure of cohomology complexes beyond prior work.

Abstract

We prove the plectic conjecture of Nekov\'a\v{r}-Scholl over global function fields $Q$. For example, when the cocharacter is defined over $Q$ and the structure group is a Weil restriction from a geometric degree $d$ separable extension $F/Q$, consider the complex computing $\ell$-adic intersection cohomology with compact support of the associated moduli space of shtukas over $Q_I$. We endow this with the structure of a complex of $\DeclareMathOperator{\Weil}{Weil}(\Weil(F)^d\rtimes\mathfrak{S}_d)^I$-modules, which extends its structure as a complex of $\Weil(Q)^I$-modules constructed by Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky. We show that the action of $(\Weil(F)^d\rtimes\mathfrak{S}_d)^I$ commutes with the Hecke action, and we give a moduli-theoretic description of the action of Frobenius elements in $\Weil(F)^{d\times I}$.