The plectic conjecture over function fields

Consider a Shimura variety whose structure group is a Weil restriction. Nekovář-Scholl conjectured that the Galois action on its étale cohomology extends to a much larger profinite group: the plectic group. After reviewing the case of Hilbert modular varieties, we present a proof of the analogue of this conjecture for moduli spaces of shtukas, which are the function-field analogue of Shimura varieties. The proof crucially uses the ability of shtukas to have multiple legs.