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

Petar Bakić, Aleksander Horawa, Siyan Daniel Li-Huerta, Naomi Sweeting

TL;DR

The paper advances the explicit construction and analysis of global long-root $A$-packets for the split group $\mathsf{G}_2$ over a number field, focusing on dihedral automorphic inductions $\tau$ from $\mathrm{PGL}_2$ via a quadratic extension $K/F$. It builds these packets through an exceptional theta lift from the unitary group $\operatorname{PU}_3$, situating the problem in the $\mathsf{G}_2\times\operatorname{PU}_3$ dual pair inside the quasi-split adjoint $\mathsf{E}_6$ and exploiting Freudenthal–Jordan algebras to connect the two sides. The authors prove Arthur’s multiplicity formula for these global $A$-packets under specific hypotheses and show that the cuspidal part forms a full near-equivalence class, thereby producing new quaternionic modular forms on $\mathsf{G}_2$. They also develop detailed local–global analyses of the local $A$-packets at finite and archimedean places, establish cuspidality criteria for the lift from $\operatorname{PU}_3$ to $\mathsf{G}_2$, and derive non-vanishing results via torus periods and Siegel–Weil type identities, tying $L$-values to the nonvanishing of the global theta lift. Overall, the work provides a concrete realization of the long-root $A$-packets for $\mathsf{G}_2$ in the dihedral case and connects automorphic, representation-theoretic, and algebraic (Freudenthal–Jordan) structures in a unified framework with potential extensions beyond the stated hypotheses.

Abstract

Cuspidal automorphic representations $τ$ 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 $τ$ 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 $\mathsf{G}_2$.

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

TL;DR

The paper advances the explicit construction and analysis of global long-root -packets for the split group over a number field, focusing on dihedral automorphic inductions from via a quadratic extension . It builds these packets through an exceptional theta lift from the unitary group , situating the problem in the dual pair inside the quasi-split adjoint and exploiting Freudenthal–Jordan algebras to connect the two sides. The authors prove Arthur’s multiplicity formula for these global -packets under specific hypotheses and show that the cuspidal part forms a full near-equivalence class, thereby producing new quaternionic modular forms on . They also develop detailed local–global analyses of the local -packets at finite and archimedean places, establish cuspidality criteria for the lift from to , and derive non-vanishing results via torus periods and Siegel–Weil type identities, tying -values to the nonvanishing of the global theta lift. Overall, the work provides a concrete realization of the long-root -packets for in the dihedral case and connects automorphic, representation-theoretic, and algebraic (Freudenthal–Jordan) structures in a unified framework with potential extensions beyond the stated hypotheses.

Abstract

Cuspidal automorphic representations of correspond to global long root -parameters for . Using an exceptional theta lift between and , we construct the associated global -packet and prove the Arthur multiplicity formula for these representations when 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 .
Paper Structure (38 sections, 62 theorems, 119 equations, 1 figure)

This paper contains 38 sections, 62 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $\mathsf{G}_2\times\operatorname{PU}_3$ theta lifts
  3. Freudenthal--Jordan algebras
  4. A quasi-split adjoint form of $\mathsf{E}_6$
  5. Dual pairs in $\widetilde{G}$
  6. Root spaces and simple roots
  7. Heisenberg parabolics
  8. Other parabolics
  9. Local theta lifts
  10. Global theta lifts
  11. Automorphic representations of $\operatorname{PU}_3$
  12. The relation with $\operatorname{U}_3$
  13. Subgroups of $\operatorname{PU}_3$
  14. The $\operatorname{U}_1\times\operatorname{U}_3$ theta lifts
  15. Local $A$-packets over $p$-adic fields
  16. ...and 23 more sections

Key Result

Theorem A

Assume that Then we have an isomorphism of $G(\mathbb A_F)$-representations where $(\epsilon_v)_v$ runs over sequences in $\{\pm1\}$ indexed by places $v$ of $F$ such that Moreover, $\bigotimes'_v\pi_v^{\epsilon_v}$ is not cuspidal if and only if every $\epsilon_v$ equals $+1$ and $L(\frac{1}{2},\chi^3)$ is nonzero.

Figures (1)

  • Figure 1: An illustration of the expected bijection between the Vogan packets $\{\sigma^+_{2,1},\sigma^-_{2,1},\sigma^-_{3,0}\}$ and $\{\pi^+,\pi^-,\pi^-_c\}$. The dotted blue arrows are the theta correspondence for $\operatorname{PU}_{2,1} \times G$ and indicate parts (1) and (3) of Conjecture \ref{['conj:G2HPSarch']}. The dashed orange arrow is the theta correspondence for $\operatorname{PU}_{2,1}\times G_c$; it would follow from Howe duality, since we know that $\theta(\pi^-_c)=\sigma^-_{2,1}$. The green arrow is the theta correspondence for $\operatorname{PU}_{3,0} \times G$ and indicates part (2) of Conjecture \ref{['conj:G2HPSarch']}, which holds by NDPC.

Theorems & Definitions (131)

  • Theorem A: Theorem \ref{['thm:actualA']}.(3)
  • Remark 1.1
  • Remark 1.2
  • Theorem B: Corollary \ref{['cor:thetacuspidal']}, Theorem \ref{['cor:non-vanishing']}
  • Theorem C: Proposition \ref{['prop:thetabackfull']}
  • Example 2.1
  • Proposition 2.2: gan2021twisted
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 121 more