The enhanced Shahidi conjecture for general quasi-split reductive groups over non-Archimedean local fields

For a quasi-split connected reductive group $G(F)$ over a non-Archimedean local field $F$, the enhanced Shahidi conjecture predicts that, within Arthur's conjectural theory of local Arthur packets, a local Arthur packet is tempered exactly when it contains a generic (Whittaker) representation; extended to all local Arthur parameters, it predicts that the packet is generic — that is, its parameter is trivial on Arthur's $\mathrm{SL}_2(\mathbb{C})$ factor — exactly when it contains a generic member. The conjecture is proved unconditionally only for split symplectic and split odd special orthogonal groups (Hazeltine–Liu–Lo, 2024, with an independent proof by Hazeltine–Liu–Lo–Zhang), for real groups (Arancibia Robert, 2025), and conditionally in several neighboring settings (Liu–Shahidi under wavefront assumptions; ABV-packet methods under the $p$-adic Kazhdan–Lusztig hypothesis). The problem is to prove both implications of both parts for arbitrary quasi-split $G$, without auxiliary hypotheses beyond the assumed local Arthur packet theory (or to refute the conjecture by an explicit counterexample).

Background

Let $F$ be a non-Archimedean (e.g. $p$-adic) local field with Weil group $W_F$, i.e. the dense subgroup of the absolute Galois group of $F$ that carries local class field theory, and let $G$ be a connected reductive group over $F$ that is quasi-split, meaning it admits a Borel subgroup (a maximal connected solvable algebraic subgroup) defined over $F$. The local Langlands program organizes the irreducible smooth complex representations of $G(F)$ by Langlands parameters, i.e. roughly speaking admissible homomorphisms $\phi:W_F\times \mathrm{SL}_2(\mathbb{C})\to{}^LG$ into the $L$-group ${}^LG$ of $G$, the semidirect product of the complex reductive dual group of $G$ with the Galois group of $F$ acting on it, up to conjugacy; the set of representations attached to one parameter should form a finite $L$-packet. Going beyond tempered representations, Arthur (1989) conjectured the existence of local Arthur packets: to each local Arthur parameter — a homomorphism of the form $\psi=\bigoplus_i \rho_i\otimes S_{a_i}\otimes S_{b_i}$, where $S_a$ denotes the $a$-dimensional irreducible algebraic representation of $\mathrm{SL}_2(\mathbb{C})$ and each $\rho_i$ is an irreducible parameter of $W_F$ of the form $\rho_i=\phi_i|\cdot|^{x_i}$, with $\phi_i$ an irreducible bounded parameter of $W_F$ and $x_i\in\mathbb{R}$ satisfying $|x_i|<\frac{1}{2}$, so that the twists $|\cdot|^{x_i}$ need not be trivial — there should correspond a finite packet $\Pi_\psi$ of irreducible smooth representations of $G(F)$, compatible with the $L$-packet of the associated Langlands parameter $\phi_\psi$ (obtained by composing $\psi$ with the diagonal map $w\mapsto\bigl(w,\mathrm{diag}(|w|^{1/2},|w|^{-1/2})\bigr)$ into Arthur's $\mathrm{SL}_2(\mathbb{C})$ factor). Write $\Psi^+(G)$ for the set of all local Arthur parameters — the class obtained by restricting global Arthur parameters to $F$, hence the parameters expected to arise from global automorphic representations — and $\Psi(G)\subseteq\Psi^+(G)$ for the subset with bounded restriction to $W_F$, that is, with $x_i=0$ for all $i$. A smooth representation is called generic if it admits a Whittaker model, i.e. a nonzero embedding into the space of functions on $G(F)$ transforming by a non-degenerate character of the unipotent radical of a Borel subgroup; genericity detects the representations to which Fourier coefficients and $L$-function theory apply most directly. A representation is tempered if its matrix coefficients decay suitably (its Langlands parameter has bounded image on $W_F$); a packet is called tempered when every member is. A local Arthur parameter $\psi$ is called generic if its restriction to Arthur's $\mathrm{SL}_2(\mathbb{C})$ factor is trivial, and the packet $\Pi_\psi$ is called generic when its parameter $\psi$ is generic. Shahidi's conjecture asserts that the tempered $L$-packets of a quasi-split group are exactly those containing a generic representation, and Shahidi (2011) proposed the enhanced version for Arthur packets: for $\psi\in\Psi(G)$, the packet $\Pi_\psi$ is tempered if and only if it has a generic member, and for $\psi\in\Psi^+(G)$, $\Pi_\psi$ is generic (that is, $\psi$ is trivial on Arthur's $\mathrm{SL}_2(\mathbb{C})$ factor) if and only if it has a generic member. This statement is central to the local Langlands program and to global applications: Hazeltine–Liu–Lo (2024) show that it implies that an automorphic representation in the discrete spectrum of $G_n(\mathbb{A}_k)$ — where $G_n$ is a split classical group $\mathrm{Sp}_{2n}$ or $\mathrm{SO}_{2n+1}$ over a number field $k$, $\mathbb{A}_k$ is its adele ring, and the discrete spectrum consists of the irreducible constituents of $L^2$ — with one generic finite local component is generic at almost all places, and, conditional on the Ramanujan conjecture for $\mathrm{GL}_n$ (the conjecture that every local component of a cuspidal automorphic representation of $\mathrm{GL}_n$ is tempered), Clozel's conjectures (2007) on the Satake parameters (the Hecke eigenvalue data attached to a representation unramified at a place) of unramified components and on the weights in the discrete spectrum. Prior to 2024 the conjecture was open in general: Liu–Shahidi proved it for symplectic and quasi-split special orthogonal groups assuming wavefront-set properties of certain bitorsor representations, and Mœglin's and Muić–Shahidi's earlier results covered special cases. Hazeltine–Liu–Lo (2024) gave the first complete unconditional proof, but only for the split classical groups $G_n=\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n+1}$, using the intersection theory of local Arthur packets and the explicit classification of generic representations of these groups. The general quasi-split case remains the outstanding target: for quasi-split unitary groups, even orthogonal groups, and non-split quasi-split forms of the classical groups, local Arthur packets have been constructed (by Arthur, Mok, and Mœglin–Waldspurger–Xu) although the conjecture is open there, whereas for exceptional groups even a construction of local Arthur packets is not yet available.

Problem Statement

Prove (or refute) the enhanced Shahidi conjecture in its stated generality: let $F$ be an arbitrary non-Archimedean local field and $G$ an arbitrary quasi-split connected reductive group over $F$, and assume a theory of local Arthur packets for $G(F)$ as conjectured by Arthur (Astérisque 171–172, 1989, §6). Then: (1) for every local Arthur parameter $\psi\in\Psi(G)$ (i.e. with bounded restriction to the Weil group $W_F$, equivalently with all twist exponents $x_i$ equal to $0$), the local Arthur packet $\Pi_\psi$ is tempered if and only if it contains a generic representation (one admitting a Whittaker model); and (2) for every local Arthur parameter $\psi\in\Psi^+(G)$ (the twist exponents $x_i$ in $\psi|_{W_F}=\bigoplus_i \phi_i|\cdot|^{x_i}$ need not vanish), the packet $\Pi_\psi$ is generic — that is, $\psi$ is trivial on Arthur's $\mathrm{SL}_2(\mathbb{C})$ factor — if and only if $\Pi_\psi$ contains a generic representation (one admitting a Whittaker model). Both implications of both parts must hold for all quasi-split $G$ without auxiliary hypotheses beyond Arthur's conjectural local Arthur packet theory assumed above; currently only the split classical groups $\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n+1}$ are settled unconditionally, the real-field analogue is known, and one direction is known for general $G$ only under the local Langlands correspondence and the $p$-adic Kazhdan–Lusztig hypothesis.

References

1. Alexander Hazeltine, Baiying Liu, Chi-Heng Lo: On the enhanced Shahidi conjecture and global applications. arXiv:2404.05773 (2024), DOI 10.48550/arXiv.2404.05773, https://arxiv.org/abs/2404.05773.
2. Freydoon Shahidi: Arthur packets and the Ramanujan conjecture. Kyoto Journal of Mathematics 51 (2011), no. 1, 1–23, DOI 10.1215/0023608X-2010-018, https://doi.org/10.1215/0023608X-2010-018.
3. Laurent Clozel: Spectral Theory of Automorphic Forms. In: Automorphic Forms and Applications, IAS/Park City Mathematics Series 12, American Mathematical Society, 2007, pp. 41–93, DOI 10.1090/pcms/012/03, https://doi.org/10.1090/pcms/012/03.
4. James Arthur: Unipotent automorphic representations: Conjectures. Astérisque 171–172 (1989), pp. 13–71, https://geodesic.mathdoc.fr/item/AST_1989__171-172__13_0/.
5. Alexander Hazeltine, Baiying Liu, Chi-Heng Lo, Qing Zhang: The closure ordering conjecture on local Arthur packets of classical groups. Journal für die reine und angewandte Mathematik (2025), DOI 10.1515/crelle-2025-0012, arXiv:2209.03816, https://doi.org/10.1515/crelle-2025-0012.
6. Baiying Liu, Freydoon Shahidi: The Jiang conjecture on the wavefront sets of local Arthur packets. arXiv:2503.05343 (2025), https://arxiv.org/abs/2503.05343.
7. Clifton Cunningham, Sarah Dijols, Andrew Fiori, Qing Zhang: Generic representations, open parameters and ABV-packets for $p$-adic groups. arXiv:2404.07463 (2024), https://arxiv.org/abs/2404.07463.
8. Kristaps Balodis, Clifton Cunningham, Andrew Fiori, Qing Zhang: Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters. arXiv:2504.04163 (2025), https://arxiv.org/abs/2504.04163.
9. Nicolas Arancibia Robert: Micro-packets containing generic representations. arXiv:2510.26343 (2025), https://arxiv.org/abs/2510.26343.