Global long root $A$-packets for $\mathsf{G}_2$: the dihedral case

TL;DR

This paper constructs and analyzes specific automorphic representations of G2 linked to PGL2 cusp forms, using theta lifts, and proves their properties including the Arthur multiplicity formula in the dihedral case.

Contribution

It introduces a new construction of global A-packets for G2 via theta lifts from PU3, proving the Arthur multiplicity formula for dihedral cases and establishing a full near equivalence class.

Findings

Constructed global A-packets for G2 using theta lifts.

Proved Arthur multiplicity formula for dihedral cases.

Identified new quaternionic modular forms on G2.

Abstract

Cuspidal automorphic representations $\tau$ of $\mathrm{PGL}_2$ correspond to global long root $A$-parameters for $\mathsf{G}_2$. Using an exceptional theta lift between $\mathrm{PU}_3$ and $\mathsf{G}_2$, we construct the associated global $A$-packet and prove the Arthur multiplicity formula for these representations when $\tau$ is dihedral and satisfies some technical hypotheses. We also prove that this subspace of the discrete automorphic spectrum forms a full near equivalence class. Our construction yields new examples of quaternionic modular forms on $\mathsf{G}_2$.