Genericity of ABV-packets for general $p$-adic reductive groups and Vogan's conjecture identifying ABV-packets with Arthur packets beyond tempered parameters

The microlocal (Adams–Barbasch–Vogan) approach to the local Langlands correspondence predicts, for every connected reductive group $G$ over a $p$-adic field, that the geometrically defined ABV-packet of a Langlands parameter coincides with the corresponding Arthur packet with matching vanishing-cycle parametrization, and that packet genericity is detected purely geometrically by openness of the parameter. As of the 2024 paper of Cunningham–Dijols–Fiori–Zhang, the genericity conjecture is proved only for quasi-split classical groups and their pure inner forms, and the ABV-packets-equal-Arthur-packets conjecture (Vogan's conjecture on A-packets) only for tempered parameters in groups where Arthur's conjectures hold. This problem asks for the genericity conjecture for general connected reductive $G$, and for Vogan's conjecture beyond tempered parameters in the scope where Arthur packets are defined by the endoscopic classification, or an explicit counterexample. Both cases remain open, with only conditional partial progress.

Background

Let $F$ be a non-archimedean local field (a $p$-adic field) and $G$ a connected reductive algebraic group over $F$, for example $\mathrm{GL}_n$, a symplectic, special orthogonal, unitary, or exceptional group. The local Langlands correspondence (LLC) predicts a finite-to-one map from irreducible admissible representations of $G(F)$ to Langlands parameters: homomorphisms $\phi$ from the Weil–Deligne group of $F$ (the Weil group of $F$ — the subgroup of the absolute Galois group consisting of elements acting as an integral power of Frobenius on the residue field, together with a nilpotent operator) into the Langlands dual group ${}^{\vee}G$, taken up to conjugacy. The fibers of this map are the $L$-packets. A representation of $G(F)$ is called generic if it admits a nontrivial Whittaker functional, i.e. a functional transformed by a non-degenerate character of a maximal unipotent subgroup; generic representations are the basic building blocks in the harmonic analysis on $G(F)$. Arthur extended the notion of packet: an Arthur parameter $\psi$ is a Langlands parameter accompanied by a further $\mathrm{SL}_2$-factor, and the predicted packet $\Pi_\psi(G)$ (an Arthur packet, or A-packet) plays a central role in the endoscopic classification of representations (carried out by Arthur for quasi-split classical groups). An Arthur parameter is tempered when its restriction to this $\mathrm{SL}_2$ factor is trivial; tempered parameters correspond to tempered representations. The adjoint $L$-function $L(s,\phi,\mathrm{Ad})$ is the Artin $L$-function of $\phi$ composed with the adjoint representation of ${}^{\vee}G$ on its Lie algebra. Gross and Prasad conjectured (1992, Conjecture 2.6, extending Gross–Prasad–Rallis) that an $L$-packet contains a generic representation if and only if $L(s,\phi,\mathrm{Ad})$ is regular (holomorphic) at $s=1$; Gan and Ichino proved this for classical groups in 2016. Vogan proposed a purely local, geometric reformulation of the LLC for all local fields. Fixing an infinitesimal parameter $\lambda$ (the restriction of $\phi$ to the Weil group of $F$), the parameters with infinitesimal parameter $\lambda$ form an affine variety $V_\lambda$ with finitely many ${}^{\vee}G$-orbits; $\phi$ is an open parameter if its orbit is open in $V_\lambda$. The Adams–Barbasch–Vogan (ABV) packet of $\phi$ is defined without any appeal to harmonic analysis: it is built by applying vanishing-cycle functors (derived functors that record how a sheaf degenerates across the fibers of a deformation, here the family of parameters with fixed infinitesimal parameter) to perverse sheaves (a class of constructible complexes of sheaves whose cohomology satisfies support conditions, well adapted to the stratification of $V_\lambda$ into ${}^{\vee}G$-orbits) on $V_\lambda$ and passing to the microlocal equivariant fundamental group $A_{\phi}^{\mathrm{ABV}}$ (a finite group playing the role of the component group of the centralizer). This construction, developed for $p$-adic groups by Cunningham–Fiori–Moussaoui–Mracek–Xu (Mem. Amer. Math. Soc. 2022), yields a packet $\Pi^{\mathrm{ABV}}_\phi(G)$ together with a canonical parametrization by characters of $A_{\phi}^{\mathrm{ABV}}$, normalized by a Whittaker datum (a choice of non-degenerate character of a unipotent radical). A theorem of Cunningham–Dijols–Fiori–Zhang (2024, building on Dat–Helm–Kurinczuk–Moss) shows $\phi$ is open if and only if $L(s,\phi,\mathrm{Ad})$ is regular at $s=1$, so the geometric openness condition matches the analytic genericity condition of Gross–Prasad. Vogan's conjecture on A-packets (appearing in Vogan's 1993 paper and made precise in the 2022 Memoir) asserts that the geometric ABV-packet machinery recovers harmonic-analytic packets: for an Arthur parameter $\psi$, the ABV-packet of the underlying Langlands parameter $\phi_\psi$ equals the Arthur packet $\Pi_\psi(G)$, and Arthur's map from the packet to characters of the component group $A_\psi$ coincides with the vanishing-cycle parametrization. The Memoir states this conjecture (Conjecture 1, Section 8.3) for quasisplit symplectic or special orthogonal $p$-adic groups, and warns that the pithy slogan "Arthur packets are ABV-packets for $p$-adic groups" obscures the fact that Arthur packets are currently defined only via Arthur's endoscopic classification, separately for each inner rational form, while ABV-packets treat all pure rational forms at once. If true in general, the conjecture would define Arthur packets for all $p$-adic reductive groups by pure geometry, freeing them from the endoscopic machine, which is currently available only for classical groups. In their 2024 paper arXiv:2412.06824, Cunningham–Dijols–Fiori–Zhang prove: (a) ABV-packets are independent of the Whittaker datum, with the parametrization's dependence made exact; (b) the genericity conjecture — an ABV-packet contains a generic representation iff $\phi$ is open (their Conjectures 4.1 and 4.3) — for quasi-split classical groups and their pure inner forms (also settling Conjecture B of Solleveld 2023 in those cases, with $G_2$ checkable from known LLC); and (c) Vogan's conjecture for tempered Arthur parameters (their Corollary 5.2), for quasi-split classical groups and any group where Arthur's conjectures hold. They also show (with their companion paper arXiv:2404.07463) that the genericity conjecture holds for any group satisfying a $p$-adic analogue of the Kazhdan–Lusztig hypothesis. What remains is the general group case of the genericity conjecture and the non-tempered case of Vogan's conjecture.

Problem Statement

Let $F$ be a non-archimedean local field. Prove or refute the two linked conjectures of the microlocal local Langlands program that arXiv:2412.06824 establishes only in special cases, each in the scope in which the sources state it.

(1) Genericity of packets from open parameters (Conjectures 4.1 and 4.3 of arXiv:2412.06824): let $G$ be an arbitrary connected reductive algebraic group over $F$ (not assumed quasi-split, classical, or exceptional-free). For every Langlands parameter $\phi$ for $G$, the pure $L$-packet $\Pi^{\mathrm{pure}}_\phi(G)$ (Conjecture 4.1) and the pure ABV-packet $\Pi^{\mathrm{ABV,pure}}_\phi(G)$ (Conjecture 4.3, stated for any connected reductive algebraic group over $F$) each contain a generic representation if and only if $\phi$ is an open parameter, i.e. its ${}^{\vee}G$-orbit is open in the variety $V_\lambda$ of Langlands parameters with infinitesimal parameter $\lambda$; equivalently (using the now-general equivalence between openness and regularity of the adjoint $L$-function) if and only if $L(s,\phi,\mathrm{Ad})$ is regular at $s=1$. This extends the Gross–Prasad genericity conjecture beyond the classical groups and $G_2$ where it is currently known.

(2) Vogan's conjecture on A-packets beyond the tempered case (Conjecture 1, Section 8.3 of Cunningham–Fiori–Moussaoui–Mracek–Xu, Mem. Amer. Math. Soc. 276 (2022)): let $G$ be a quasi-split classical $p$-adic group or any group for which Arthur's conjecture is known, with pure inner forms covered through the corresponding pure packets — this is the operative scope of arXiv:2412.06824, which contains the Memoir's original scope of quasisplit symplectic or special orthogonal $p$-adic groups — and let $\psi$ be an Arthur parameter of $G$ that need not be tempered. Prove that the pure Arthur packet equals the pure ABV-packet of the underlying Langlands parameter, $\Pi^{\mathrm{pure}}_\psi(G)=\Pi^{\mathrm{ABV,pure}}_{\phi_\psi}(G)$, and that Arthur's parametrization $A(\mathfrak{w})$ of the packet by characters of the component group $A_\psi$ coincides with the vanishing-cycle parametrization $\mathrm{NEvs}_{\phi_\psi}\circ \mathcal{P}_{\mathfrak{w}}$ built from normalized vanishing cycles on $V_\lambda$ (where $\mathcal{P}_{\mathfrak{w}}$ denotes the Whittaker-normalized bijection sending each representation $\pi$ in the packet to a simple perverse sheaf on $V_\lambda$, and $\mathrm{NEvs}$ the normalized-eversion functor of Cunningham–Fiori–Moussaoui–Mracek–Xu).

The common objective uniting both components is to establish Vogan's purely geometric, microlocal formulation of the local Langlands correspondence in the scopes stated above: Arthur packets and their generic members would then be defined and detectable by the geometry of the variety of Langlands parameters alone, with no appeal to endoscopic classification. Each conjecture is settled by a proof or refuted by an explicit group $G$ and parameter for which the predicted equivalence or equality of packets fails. The four sub-claims — Conjecture 4.1, Conjecture 4.3, the equality of packets $\Pi^{\mathrm{pure}}_\psi(G)=\Pi^{\mathrm{ABV,pure}}_{\phi_\psi}(G)$, and the coincidence of the two parametrizations — are linked by implication rather than independent: in the shared scope Conjecture 4.3 implies Conjecture 4.1, and the coincidence of the parametrizations presupposes the equality of the packets, so resolving a stronger sub-claim resolves the weaker one it entails and a counterexample to the weaker one refutes the stronger. Component (1) is not restricted to quasi-split or classical groups, and component (2) is not restricted to tempered parameters; for groups beyond the reach of Arthur's endoscopic classification, where $\Pi_\psi(G)$ and Arthur's map $A(\mathfrak{w})$ are not currently defined, the geometric extension of Arthur packets is a motivating aspiration rather than part of the stated objective.

References

1. Clifton Cunningham, Sarah Dijols, Andrew Fiori, Qing Zhang, 'Whittaker normalization of $p$-adic ABV-packets and Vogan's conjecture for tempered representations' (2024), arXiv:2412.06824, https://arxiv.org/abs/2412.06824
2. Clifton Cunningham, Sarah Dijols, Andrew Fiori, Qing Zhang, 'Generic representations, open parameters and ABV-packets for $p$-adic groups' (2024), arXiv:2404.07463, https://arxiv.org/abs/2404.07463
3. Kristaps Balodis, Clifton Cunningham, Andrew Fiori, Qing Zhang, 'Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters' (2025), arXiv:2504.04163, https://arxiv.org/abs/2504.04163
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