Temperedness of locally generic cuspidal automorphic representations: the global-to-local A-packet, rigidity, and local-global genericity conjectures underlying Shahidi's conditional Ramanujan theorem

For a quasisplit connected reductive group $G$ over a number field $k$, Shahidi (Kyoto J. Math. 51, 2011) proved conditionally, in his Corollary 6.5, that every locally generic cuspidal automorphic representation $\pi=\otimes_v\pi_v$ of $G(\mathbb{A}_k)$ is tempered, assuming two open conjectures: the Arthur–Clozel global-to-local conjecture that for almost all finite primes $v$ the component $\pi_v$ lies in the L-packet of the Langlands parameter underlying the local Arthur parameter (Conjecture 6.1), and the rigidity conjecture that temperedness at almost all — or even just infinitely many — places forces temperedness at every place (Conjecture 6.4); the same paper also poses, as a companion open question, the conjecture that locally generic cuspidal representations are globally generic (Conjecture 2.10). The later literature settles parts of this circle: over function fields a rigidity-type theorem is now unconditional (Ciubotaru–Harris), and for symplectic and split odd orthogonal groups the global-to-local statement follows from Arthur's endoscopic classification (Hazeltine–Liu–Lo). Over number fields, however, the rigidity and local–global genericity conjectures remain open in general, the global-to-local conjecture is open beyond the endoscopic-classification cases, and the head objective implies the still-open generalized Ramanujan conjecture for $\mathrm{GL}_N$.

Background

An automorphic representation of a connected reductive group $G$ over a number field $k$ is an irreducible admissible representation $\pi=\otimes_v \pi_v$ of the adelic points $G(\mathbb{A}_k)$ occurring in the space of square-integrable automorphic functions; it is cuspidal when it occurs in the discrete part of the spectrum built from cusp forms, i.e. functions on $G(k)\backslash G(\mathbb{A}_k)$ that are orthogonal to all constant terms along proper parabolic subgroups. The group $G$ is called quasisplit if it has a Borel subgroup defined over $k$. An irreducible admissible representation of $G(k_v)$ is tempered if its matrix coefficients lie in $L^{2+\varepsilon}$ for every $\varepsilon>0$, equivalently if its Langlands parameter is bounded and it is (a limit of) a representation unitarily induced from discrete series — temperedness is the representation-theoretic expression of 'no complementary series' (no unitary representations continuously interpolating away from the unitary dual). A representation is generic if it admits a Whittaker model, i.e. if its space realizes nonzero functions on $G(k_v)$ transforming by a non-degenerate character of a maximal unipotent subgroup; a cuspidal automorphic representation is globally generic if the analogous global Whittaker integral is nonzero on it, and locally generic if each local component $\pi_v$ is generic. Global genericity implies local genericity; whether the converse holds is itself open (Conjecture 2.10 below). The Langlands program predicts that irreducible representations of $G(k_v)$ are parametrized by Langlands parameters, homomorphisms $\phi: L_{k_v} \to {}^L G$ from the Weil–Deligne group $L_{k_v}$ (the object packaging the local absolute Galois group together with a nilpotent operator, used to classify representations of $G(k_v)$) into the L-group ${}^L G$ (the semidirect product of the complex dual group of $G$ with a Weil group), finite sets of which form L-packets. Arthur extended this picture: an Arthur parameter $\psi=(\phi,\rho)$ pairs a Langlands parameter with a representation $\rho$ of an auxiliary $\mathrm{SL}_2(\mathbb{C})$, and the conjectural A-packet $\Pi(\psi_v)$ attached to $\psi_v$ contains the possible local components of global automorphic representations with global Arthur parameter $\psi$; the subpacket $\Pi(\phi_{\psi_v})$ attached to the Langlands parameter $\phi_{\psi_v}$ underlying $\psi_v$ is expected to capture the 'tempered part'. The classical Ramanujan conjecture for $\mathrm{GL}_N$, proved over function fields by L. Lafforgue but open over number fields (already for Maass forms, i.e. non-holomorphic cuspidal eigenfunctions of the hyperbolic Laplacian, on $\mathrm{GL}_2$ over $\mathbb{Q}$), states that every cuspidal automorphic representation of $\mathrm{GL}_N(\mathbb{A}_k)$ with unitary central character is tempered at every place. For groups other than $\mathrm{GL}_N$ the literal statement fails: there exist non-tempered cuspidal representations, such as the Saito–Kurokawa lifts on $\mathrm{Sp}_4$ (cuspidal representations of the rank-two symplectic group, lifted from automorphic forms of lower rank, that fail to be tempered) and, more generally, CAP representations (Cuspidal Representations Associated to Parabolics), which are locally non-tempered almost everywhere. The correct generalization, going back to Howe–Piatetski-Shapiro and formulated precisely in the Langlands–Arthur framework, is that the generic spectrum should be tempered. Shahidi's 2011 paper made the dependence of this statement on the Arthur conjectures explicit: Theorem 6.2 shows, assuming the global-to-local conjecture (Conjecture 6.1 below), that a locally generic cuspidal $\pi$ is tempered at almost all places; a purely local ingredient (his Theorem 5.1) requires the local Langlands correspondence for every proper Levi subgroup $M$ of $G$, in the form of the equality $L(s, r\cdot\phi_\nu) = L(s,\sigma_\nu, \widetilde r)$ between the Artin $L$-factor of the parameter $\phi_\nu$ (the factor attached to $\phi_\nu$ viewed as a representation of the Weil–Deligne group) and the Langlands–Shahidi $L$-function (the local factor defined by the Shahidi method through generic induced representations and their Whittaker functionals) for generic tempered representations $\sigma_\nu$ of $M(k)$, where $r$ is the adjoint action of ${}^L M$ on the Lie algebra of the L-group of the unipotent radical. Shahidi's Corollary 6.5 (statement (2.12) of the paper) derives the temperedness statement from the global-to-local conjecture (Conjecture 6.1) together with the rigidity conjecture (Conjecture 6.4), the latter needed only in its locally generic instance. The paper is explicit that its supporting conjectures, and hence its conclusion that locally generic cuspidal representations are tempered everywhere (Corollary 6.5), remain unproved; indeed Corollary 6.5 implies the Ramanujan conjecture for $\mathrm{GL}_N$, since cuspidal representations of $\mathrm{GL}_N$ are generic, and as recorded by Ciubotaru–Harris (after a remark of Wee Teck Gan communicated by Clozel), the version of the Arthur conjectures assumed in Theorem 6.2 already includes Ramanujan for $\mathrm{GL}_N$.

Problem Statement

For $G$ a quasisplit connected reductive group over a number field $k$, establish unconditionally that every locally generic cuspidal automorphic representation $\pi=\otimes_v\pi_v$ of $G(\mathbb{A}_k)$ (with unitary central character) is tempered at every place of $k$. The following three conjectures are all stated in that paper; the third is a companion open conjecture posed in the same work and included here as part of the paper's open circle of questions:

1. Global-to-local A-packet conjecture (Conjecture 6.1, Arthur; Clozel's Conjecture 2A): for almost all finite primes $v$ of $k$, the local component $\pi_v$ belongs to the L-packet $\Pi(\phi_{\psi_v})$ attached to the Langlands parameter $\phi_{\psi_v}$ underlying the local Arthur parameter $\psi_v$ of $\pi_v$.

2. Rigidity conjecture (Conjecture 6.4): every cuspidal automorphic representation $\pi=\otimes_v\pi_v$ of $G(\mathbb{A}_k)$ whose local components $\pi_v$ are tempered for almost all places $v$ — or even merely for infinitely many places — is tempered, i.e. $\pi_v$ is tempered for every place $v$. (As stated in the source, this ranges over all cuspidal automorphic representations, not only locally generic ones.)

3. Local-global genericity (Conjecture 2.10): if $\pi=\otimes_v\pi_v$ is a cuspidal automorphic representation that is locally generic with respect to local components of a generic character $\chi=\otimes_v\chi_v$ of $U(k)\backslash U(\mathbb{A}_k)$, where $U$ is a maximal unipotent subgroup of $G$, then the isotypic constituent of the cuspidal spectrum containing $\pi$ is globally generic with respect to $\chi$ (the global Whittaker integral is nonzero on it).

Conjecture 3 is not a hypothesis of the temperedness statement but a consequence of it: the temperedness statement is what removes the known global obstructions (conditions on adjoint $L$-functions) to passing from local to global genericity, which is why the paper poses Conjecture 2.10 alongside the other two; consequently a negative answer to Conjecture 3 (a locally generic cuspidal representation that is not globally generic with respect to any global generic character) would itself be an explicit locally generic cuspidal representation that fails to be tempered, and would therefore refute the temperedness objective. Conversely, the statement for $G=\mathrm{GL}_N$ is exactly the Ramanujan conjecture for $\mathrm{GL}_N$ (Shahidi's remark (6.7)), since cuspidal representations of $\mathrm{GL}_N$ are generic, so a proof at the stated generality subsumes the generalized Ramanujan conjecture for $\mathrm{GL}_N$ over number fields. The problem is answered in full by an unconditional proof for all quasisplit $G$, and is refuted by an explicit locally generic cuspidal representation that fails to be tempered. A counterexample to the bundled rigidity conjecture (item 2) — a cuspidal representation tempered at infinitely many places with an explicit non-tempered local component — would not by itself refute the temperedness objective, since rigidity is a sufficient ingredient of Corollary 6.5 rather than an equivalent formulation of it. The task is resolved exactly when the temperedness statement above is proved or refuted.

References

1. Freydoon Shahidi, "Arthur packets and the Ramanujan conjecture", Kyoto Journal of Mathematics 51 (2011), 1–23. DOI: 10.1215/0023608X-2010-018; arXiv:1007.2132. URL: https://doi.org/10.1215/0023608X-2010-018
2. Laurent Clozel, "Spectral theory of automorphic forms", in Automorphic Forms and Applications (Park City, 2002), IAS/Park City Mathematics Series 12, American Mathematical Society, 2007, pp. 43–93. DOI: 10.1090/pcms/012/03. URL: https://doi.org/10.1090/pcms/012/03
3. James Arthur, The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups, American Mathematical Society Colloquium Publications 61, Providence, RI, 2013. ISBN: 978-0-8218-4990-3. DOI: 10.1090/coll/061. URL: https://doi.org/10.1090/coll/061
4. Chung Pang Mok, "Endoscopic classification of representations of quasi-split unitary groups", Memoirs of the American Mathematical Society 1108 (2015). DOI: 10.1090/memo/1108. URL: https://doi.org/10.1090/memo/1108
5. Dan Ciubotaru and Michael Harris, "On the generalized Ramanujan conjecture over function fields", preprint (2022). arXiv:2204.06053. URL: https://arxiv.org/abs/2204.06053
6. Dan Ciubotaru and Michael Harris, "On the generalized Ramanujan and Arthur conjectures over function fields", preprint (2023). arXiv:2311.15300. URL: https://arxiv.org/abs/2311.15300
7. Alexander Hazeltine, Baiying Liu, and Chi-Heng Lo, "On the enhanced Shahidi conjecture and global applications", preprint (2024). arXiv:2404.05773. URL: https://arxiv.org/abs/2404.05773
8. Yeansu Kim, Muthu Krishnamurthy, and Freydoon Shahidi, "L-packets and the generic Arthur packet conjectures for even unitary similitude groups", preprint (2025). arXiv:2506.00892. URL: https://arxiv.org/abs/2506.00892
9. Baiying Liu and Freydoon Shahidi, "On Jiang's wavefront sets conjecture for representations in local Arthur packets", preprint (2026). arXiv:2603.13465. URL: https://arxiv.org/abs/2603.13465