Construction of Local Arthur Packets for Metaplectic Groups and the Adams Conjecture
Pith reviewed 2026-05-15 12:24 UTC · model grok-4.3
The pith
Local Arthur packets for metaplectic groups over non-Archimedean fields are constructed explicitly by generalizing the classical groups case, which establishes multiplicity freeness and extends the Adams conjecture.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We explicitly construct local Arthur packets for metaplectic groups over non-Archimedean local fields of characteristic zero. Our construction is a generalization of Atobe's construction of local Arthur packets for classical groups. As a result, we prove that the local Arthur packets are multiplicity free. Moreover, we generalize Moeglin's earlier work about the Adams conjecture to metaplectic groups.
What carries the argument
The generalized explicit construction of local Arthur packets, obtained by extending Atobe's method to assign to each Arthur parameter a finite set of irreducible representations of the metaplectic group.
If this is right
- The local Arthur packets attached to metaplectic groups are multiplicity-free.
- Moeglin's formulation of the Adams conjecture holds for metaplectic groups.
- Every Arthur parameter for a metaplectic group receives a well-defined finite packet of irreducible representations.
- The endoscopic classification of representations extends to metaplectic groups via these packets.
Where Pith is reading between the lines
- The construction supplies a concrete model that could be used to check compatibility between local packets and global automorphic forms on metaplectic covers.
- Multiplicity-freeness may simplify the calculation of global multiplicities when metaplectic groups appear in endoscopic transfers.
- The same generalization technique might apply to other nonlinear covers once the classical-group case is fully understood.
Load-bearing premise
The proofs and verifications developed for classical groups extend directly to metaplectic groups without new obstructions or extra cases to check.
What would settle it
An explicit Arthur parameter for a small metaplectic group (such as the double cover of SL(2)) for which the constructed packet contains at least one irreducible representation with multiplicity greater than one.
read the original abstract
In this article, we explicitly construct local Arthur packets for metaplectic groups over non-Archimedean local fields of characteristic zero. Our construction is a generalization of Atobe's construction of local Arthur packets for classical groups. As a result, we prove that the local Arthur packets are multiplicity free. Moreover, we generalize Moeglin's earlier work about the Adams conjecture to metaplectic groups.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper explicitly constructs local Arthur packets for metaplectic groups over non-Archimedean local fields of characteristic zero by generalizing Atobe's construction for classical groups. This yields a proof that the resulting packets are multiplicity-free and extends Moeglin's work on the Adams conjecture to the metaplectic setting.
Significance. If the central claims hold, the work supplies an explicit endoscopic construction of local Arthur packets for a nontrivial central extension, advancing the local Langlands correspondence for metaplectic groups. The multiplicity-free result and the Adams-conjecture generalization would be useful for global automorphic forms and for comparing genuine representations with their classical counterparts.
major comments (2)
- [§3.2] §3.2, Construction of the packet map: the argument that the endoscopic character matching lifts bijectively to the metaplectic cover relies on the same stable-distribution identity used by Atobe, but the 2-cocycle on the Weil-Deligne group is not shown to preserve the required sign relations; without an explicit verification that no extra signs appear in the genuine case, the multiplicity-free claim does not follow automatically.
- [§5] §5, Generalization of the Adams conjecture: the reduction to Moeglin's earlier result assumes that the Arthur parameter for the metaplectic group satisfies the same endoscopic transfer conditions as in the classical case, yet the paper does not supply a separate check that the cocycle obstruction vanishes for the relevant unramified parameters.
minor comments (2)
- [Introduction] Notation for the metaplectic 2-cocycle is introduced only in §2; moving a brief definition to the introduction would improve readability.
- [§3] Several references to Atobe's theorems are given without page or equation numbers; adding precise citations would help readers trace the generalizations.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments. We address the major points below and will revise the manuscript accordingly to make the arguments fully explicit.
read point-by-point responses
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Referee: [§3.2] §3.2, Construction of the packet map: the argument that the endoscopic character matching lifts bijectively to the metaplectic cover relies on the same stable-distribution identity used by Atobe, but the 2-cocycle on the Weil-Deligne group is not shown to preserve the required sign relations; without an explicit verification that no extra signs appear in the genuine case, the multiplicity-free claim does not follow automatically.
Authors: We thank the referee for highlighting this subtlety. The 2-cocycle in our construction is the natural lift of the one appearing in Atobe's work for the classical groups, chosen so that it is compatible with the central extension and factors through the projection map in a manner that preserves the sign relations on the relevant Weil-Deligne elements. Consequently, the stable-distribution identity lifts without introducing extra signs for genuine representations, and the multiplicity-free property follows as stated. To address the referee's concern directly, we will add an explicit lemma in the revised §3.2 verifying the sign preservation by direct computation on generators of the Weil-Deligne group. revision: yes
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Referee: [§5] §5, Generalization of the Adams conjecture: the reduction to Moeglin's earlier result assumes that the Arthur parameter for the metaplectic group satisfies the same endoscopic transfer conditions as in the classical case, yet the paper does not supply a separate check that the cocycle obstruction vanishes for the relevant unramified parameters.
Authors: We appreciate the referee's observation on this reduction step. For the unramified Arthur parameters under consideration, the cocycle is unramified and the metaplectic cover splits over the corresponding unramified tori and parahoric subgroups; thus the obstruction vanishes identically by the same local class-field-theoretic argument used by Moeglin. This allows the direct reduction. Nevertheless, we agree that an explicit verification would improve clarity, and we will insert a short paragraph in the revised §5 confirming that the cocycle obstruction is trivial for these unramified parameters. revision: yes
Circularity Check
No circularity; explicit generalization of external construction
full rationale
The paper presents an explicit construction of local Arthur packets for metaplectic groups as a direct generalization of Atobe's prior work on classical groups, with the multiplicity-free property and Adams conjecture extension presented as consequences. No equations or steps reduce by definition to their own inputs, no parameters are fitted and then relabeled as predictions, and no load-bearing self-citation chain is invoked. The derivation relies on external results (Atobe, Moeglin) treated as independent inputs rather than self-referential premises.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Atobe's construction for classical groups extends to metaplectic groups without new obstructions
discussion (0)
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