On the multiplicity formula for discrete automorphic representations of disconnected tori

Kaletha extended the local Langlands conjectures to disconnected groups that are inner forms of semidirect products $G\rtimes A$, where the finite group scheme $A$ preserves a pinning of the connected reductive group $G$, and proved the conjectures when $G$ is a torus.

Our first main result is an intrinsic reinterpretation of the local Langlands correspondence for such disconnected tori.

Our second main result, and the central objective of this paper, is to establish an automorphic multiplicity formula for them.