Weak Arthur Packets in p-adic Groups

• Weak Arthur packets are defined as collections of irreducible admissible p-adic group representations determined by geometric and microlocal invariants linked to an Arthur parameter.
• They decompose into unions of genuine Arthur packets, with their structure shaped by special pieces, inertia, and Barbasch–Vogan duality.
• These packets are central to the local Langlands program, connecting perverse sheaf theory, wavefront sets, and representation theory in p-adic groups.

A weak Arthur packet is a collection of irreducible admissible representations of a reductive $p$-adic group, defined with respect to an Arthur parameter, which is constructed to capture essential geometric and representation-theoretic properties associated to unipotent and, more generally, Arthur-type parameters. Unlike strong (or genuine) Arthur packets, which are determined by endoscopic transfer and internal character identities, weak Arthur packets are defined by geometric and microlocal invariants—such as wavefront sets, perverse sheaves, or characteristic cycles—and are often conjectured or shown to be unions of genuine Arthur packets. The concept, recently formalized and systematically developed for $p$-adic groups, plays a crucial role in the unipotent representation theory of classical and exceptional groups, and in the geometric realization of the local Langlands correspondence.

1. Formal Definitions and Constructions

The notion of a weak Arthur packet can be approached from several interlocking perspectives: wavefront geometry, perverse sheaf microlocality, and parameter-theoretic combinatorics. For a split classical $p$-adic group $G$, an Arthur parameter

$$\psi: W_F \times \mathrm{SL}_2(\mathbb{C})_D \times \mathrm{SL}_2(\mathbb{C})_A \to {}^L G$$

is a continuous homomorphism, bounded and algebraic on the respective factors. The "basic" or unipotent case—$\psi$ trivial on $W_F$—corresponds to nilpotent orbits in the Langlands dual Lie algebra (Liu et al., 2023, Gurevich et al., 2024).

The weak Arthur packet attached to a given $\psi$ is typically defined in terms of the leading nilpotent invariant of its members. The archetype: $$\Pi_\psi^{\mathrm{Weak}} = \left\{ \pi \in \operatorname{Irr}(G) : \mathrm{WF}(\pi) \leq d_{BV}(\mathcal{O}_\psi^D) \right\}$$ where $\mathrm{WF}(\pi)$ denotes the geometric wavefront set and $d_{BV}$ is the Barbasch–Vogan duality (a bijection between nilpotent orbits in the dual groups) [(Liu et al., 2023); §1]. In practice, for fixed infinitesimal character or unipotent class, weak packets are characterized as maximal unions of Arthur packets subject to geometric or microlocal constraints (Gurevich et al., 2024).

For exceptional groups such as $G_2$, weak Arthur (or ABV-) packets are constructed via a geometric bijection between simple $G^\vee$-equivariant perverse sheaves and irreducible unipotent representations, with the microlocal functor $\mu$ mapping to representations of a component group $A_\psi$ and the packet defined by microlocal nonvanishing: $$\Pi_\psi = \left\{ \pi : \mu_{C_\psi} P(\pi) \neq 0 \right\}$$ [(Cunningham et al., 2020), §3].

In recent work for exceptional $F_4$ groups, weak packets are constructed as Aubert–Zelevinsky duals of unipotent representations attached to orbits in the special piece of the relevant nilpotent space, with a precise correspondence to microlocal packets defined via characteristic cycles of simple perverse sheaves [(Barchini et al., 7 Dec 2025), Def. 3.1.1, Thm. 4.1].

2. Decomposition into Genuine Arthur Packets

A central conjecture, validated in numerous cases, is that weak Arthur packets are unions of (strong) Arthur packets, each distinguished by infinitesimal character and ramification properties: $$\Pi_\psi^{\mathrm{Weak}} = \bigcup_{\substack{\psi' \in \Psi(G)_X \ d_{BV}(\mathcal{O}_{\psi'}^D) = d_{BV}(\mathcal{O}_\psi^D)}} \Pi_{\psi'}$$ [(Liu et al., 2023), §2]. This structural decomposition is governed by the "special piece" stratification of the unipotent variety in the dual group, reflecting both the Barbasch–Vogan duality and the geometry of endoscopic transfer [(Gurevich et al., 2024), Thm.]. For $G_2$ and $F_4$, analogues manifest as unions of microlocal packets labeled by orbit data (Cunningham et al., 2020, Barchini et al., 7 Dec 2025).

In split classical and exceptional cases, this conjecture was proved under mild technical assumptions (notably on residual characteristic and purity), confirming that weak packets partition precisely according to special pieces and ramification classes.

3. Microlocal and Geometric Realizations

The microlocal perspective on weak packets leverages the theory of perverse sheaves, characteristic cycles, and vanishing cycles on parameter varieties. For $G_2$, the equivalence between perverse sheaf categories and unipotent representations enables the construction of weak packets via the microlocal functor and associated trace formulae (Cunningham et al., 2020):

• Simple $G_2$-equivariant perverse sheaves are classified by $GL_2$-orbits and their equivariant local systems on a four-dimensional parameter space of cubics.
• The microlocal vanishing-cycles functor $\mu$ sends perverse sheaves to representations of $A_\psi$, and the trace on this group yields the pairing $$\langle a, \pi \rangle_\psi = \operatorname{Trace}_a(\mu_{C_\psi} P(\pi))$$.
• Microlocal packets correspond to collections of representations whose geometric or microlocal multiplicities are nonzero with respect to certain orbit data.

For $F_4$, characteristic cycles $CC(IC(S, \mathcal{L}))$ are explicitly computed, and weak packets are shown to be unions of microlocal packets over special pieces of the nilpotent cone (Barchini et al., 7 Dec 2025).

4. Ramification, Sphericity, and Canonical Quotients

The composition of weak Arthur packets is closely controlled by the ramification behavior of the underlying parameters. Specifically, in symplectic and orthogonal groups, membership in a weak packet is characterized by restriction properties on inertia and the existence of compact-fixed vectors ("weak sphericity") [(Gurevich et al., 2024), §3–4]. Fundamental results establish that:

• An anti-tempered Arthur packet belongs to a weak packet if and only if the parameter is unramified on inertia and the packet contains a weakly-spherical representation.
• Canonical algebraic quotients $$A^\dagger(\mathcal{O}^\vee) \leq \widehat{A(\mathcal{O}^\vee)}$$ derived via Lusztig's theory govern which local systems and thus which representations appear in weakly-spherical (and hence in weak) packets.
• Springer correspondences and special piece partitions structure the fine geometry of the decomposition.

This provides a direct link between ramification invariants, special piece geometry, and the spectrum of weak Arthur packets.

5. Unitarity, Genericity, and Extensions

Membership in a weak Arthur packet often implies important representation-theoretic properties:

• All members of weak packets are unitary, as they are unions of unitary Arthur packets [(Liu et al., 2023), Thm. 4.7].
• Under suitable boundedness or wavefront set control (Jiang's conjectures), the weak packet precisely matches the maximal union of genuine packets with fixed leading nilpotent orbit.
• In unitary similitude groups, the "weak generic Arthur packet conjecture" establishes that only tempered parameters can give rise to packets containing generic (Whittaker) elements, and equality of $L$-functions suffices to determine the packet structure (Kim et al., 1 Jun 2025).

Furthermore, the framework generalizes to all Arthur-type representations of fixed infinitesimal character, not only unipotent representations [(Liu et al., 2023), §5].

6. Examples and Explicit Computations

Table: Explicit constructions of weak Arthur packets in select groups.

Group/Type	Packet Construction Approach	Main Decomposition Result
$G_2$	Perverse sheaves, microlocal cycles	Weak packets by microlocal nonvanishing; union of strong packets indexed by $S_3$ (Cunningham et al., 2020)
$F_4$ (minimal $\mathcal{O}$)	Characteristic cycles, AZ duality	Weak packet is union of five microlocal packets in special piece (Barchini et al., 7 Dec 2025)
Split classical (Sp/ SO)	Wavefront set geometry, BV duality	Weak packet as union over Arthur packets indexed by special piece (Gurevich et al., 2024, Liu et al., 2023)

In low rank ($SO_5$ or $Sp_4$), explicit computations show how the weak packet consists of both the basic packet and certain endoscopic packets, with parametrization governed by the fiber structure of duality maps and component groups.

7. Future Directions and Open Conjectures

Ongoing research aims to refine the relationship between geometric invariants (perverse sheaf theory, microlocal multiplicity) and the analytic theory (endoscopy, trace formulas) to fully upgrade weak Arthur packets to genuine packets with all expected transfer and stability properties. For $G_2$, current results establish conditional stability of weak packets, with future work to prove full endoscopic characterization (Cunningham et al., 2020). For broader classes of reductive groups, especially in exceptional or bad-characteristic settings, efforts focus on confirming maximal union properties, understanding ramifications of Lusztig’s quotient structure, and extending the definitions beyond unipotent types.

Given their geometric naturality and connection to the local Langlands program, weak Arthur packets are expected to remain a central object in the classification of admissible representations of $p$-adic groups and in the microlocal analysis of perverse sheaves on nilpotent parameter spaces.

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References:

• "Arthur packets for $G_2$ and perverse sheaves on cubics" (Cunningham et al., 2020)
• "Some unipotent Arthur packets for p-adic split F4" (Barchini et al., 7 Dec 2025)
• "Ramification of weak Arthur packets for p-adic groups" (Gurevich et al., 2024)
• "On the weak local Arthur packets conjecture for split classical groups" (Liu et al., 2023)
• "$L$-packets and the generic Arthur packet conjectures for even unitary similitude groups" (Kim et al., 1 Jun 2025)