Arthur Packets Research Articles

Equivalent Definitions of Arthur Packets for Real Classical Groups

Arthur has conjectured the existence of what are now known as Arthur packets of representations of reductive algebraic groups over local and global fields. In the case of special orthogonal and symplectic groups, he subsequently gave a definition of these packets, using local and global methods. For general real groups, an alternative approach to the definition of Arthur packets has been given by Adams-Barbasch-Vogan. This construction is purely local and uses geometric methods. Our main result is that these two definitions agree in the case of real special orthogonal and symplectic groups.

On the Weak Local Arthur Packets Conjecture for Split Classical Groups

Abstract Recently, motivated by the theory of real local Arthur packets, making use of the wavefront sets of representations over non-Archimedean local fields $F$, Ciubotaru, Mason-Brown, and Okada defined the weak local Arthur packets consisting of certain unipotent representations and conjectured that they are unions of local Arthur packets. In this paper, we prove this conjecture for $\textrm{Sp}_{2n}(F)$ and split $\textrm{SO}_{2n+1}(F)$ with the assumption of the residue field characteristic of $F$ being large. In particular, this implies the unitarity of these unipotent representations. We also discuss the generalization of the weak local Arthur packets beyond unipotent representations, which reveals the close connection with a conjecture of Jiang on the structure of wavefront sets for representations in local Arthur packets.

On wavefront sets of global Arthur packets of classical groups: Upper bound

We prove a conjecture of the first named author (2014) on the upper bound Fourier coefficients of automorphic forms in Arthur packets of all classical groups over any number field. This conjecture generalizes the global version of the local tempered L -packet conjecture of Shahidi (1990). Under certain assumption, we also compute the wavefront sets of the unramified unitary dual for split classical groups.

Arthur Packets for Quasisplit GSp(2n) and GO(2n) Over a p-Adic Field

Abstract We construct the Arthur packets for symplectic and even orthogonal similitude groups over a $p$-adic field and show that they are stable and satisfy the twisted endoscopic character relations.

Some Unipotent Arthur Packets for Reductive p-adic Groups

Abstract Let ${\textsf k}$ be a $p$-adic field, and let $\textbf {G}({\textsf k})$ be the ${\textsf k}$-points of a connected reductive group, inner to split. The set of Aubert–Zelevinsky duals of the constituents of a tempered L-packet forms an Arthur packet for $\textbf {G}({\textsf k})$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang–Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets, which we call weak Arthur packets and describe their constituents in terms of their Langlands parameters.

Proof of Vogan’s conjecture on Arthur packets: irreducible parameters of p-adic general linear groups

In this paper we prove Vogan’s conjecture on local Arthur packets, as recalled in Cunningham et al. [Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples, Memoirs of the American Mathematical Society, Boston, 2022, Section 8.3, Conjecture 1(a)], for irreducible Arthur parameters of p-adic general linear groups. This result shows that these Arthur packets may be characterized by properties of simple perverse sheaves on a moduli space of Langlands parameters.

On Arthur's unitarity conjecture for split real groups

Arthur's conjectures predict the existence of some very interesting unitary representations occurring in spaces of automorphic forms. We prove the unitarity of the Langlands element (i.e., the one specified by Arthur) of all unipotent Arthur packets for split real groups. The proof uses Eisenstein series, Langlands' constant term formula and square integrability criterion, analytic properties of intertwining operators, and some mild arithmetic input from the theory of Dirichlet L-functions, to reduce to a more combinatorial problem about intertwining operators.

Arthur packets for 𝑝-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

In this article we propose a geometric description of Arthur packets for p p -adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p p -adic groups and propose a geometric description of the transfer coefficients that appear in Arthur’s main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan’s work, including the prediction that Arthur packets are ABV-packets for p p -adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

Twisted composition algebras and Arthur packets for triality $\operatorname{Spin}_8$

Twisted composition algebras and Arthur packets for triality $\operatorname{Spin}_8$

Arthur packets for G2 and perverse sheaves on cubics

This paper begins the project of defining Arthur packets of all unipotent representations for the p-adic exceptional group G2. Here we treat the most interesting case by defining and computing Arthur packets with component group S3. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. We treat all other unipotent representations of p-adic G2 in forthcoming work.

On unitarizability and Arthur packets

In this paper we begin to explore the relation between the question of unitarizability of classical p-adic groups, and Arthur packets, starting from Tadić (Mem Am Math Soc 2020, to appear).

Branching laws for discrete series of some affine symmetric spaces

In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $\Pi$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representation $\Pi$ and the corresponding Arthur-Vogan packets to smaller orthogonal groups; these are inspired by the Gross-Prasad conjectures.

Branching laws for classical groups: the non-tempered case

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

Sur les paquets d'Arthur des groupes classiques réels

This article is part of a project started in [AMR18] which consists of investigating Arthur packets for real classical groups. Our goal is to give an explicit description of these packets and to establish the multiplicity one property (which is known to hold for p -adic and complex groups). The main result in this paper is a construction of packets from unipotent packets on c -Levi factors using cohomological induction. An important tool used in the argument is a result on commutativity between cohomological induction and spectral endoscopic transfer.

On the non-vanishing of the central value of certain $L$-functions: unitary groups

Let \pi be an irreducible cuspidal automorphic representation of a quasi-split unitary group U _\mathfrak n defined over a number field F . Under the assumption that \pi has a generic global Arthur parameter, we establish the non-vanishing of the central value of L -functions, L(1/2,\pi\times\chi) , with a certain automorphic character \chi of U _1 for the case of n=2,3,4 , and for the general n \geq 5 by assuming Conjecture 1.4 on certain refined properties of global Arthur packets. As consequence, we obtain some simultaneous non-vanishing results for the central L -values by means of the theory of endoscopy.

A COMBINATORIAL SOLUTION TO MŒGLIN’S PARAMETRIZATION OF ARTHUR PACKETS FOR -ADIC QUASISPLIT AND

AbstractWe develop a general procedure to study the combinatorial structure of Arthur packets for $p$-adic quasisplit $\mathit{Sp}(N)$ and $O(N)$ following the works of Mœglin. This will allow us to answer many delicate questions concerning the Arthur packets of these groups, for example the size of the packets.

On Mœglin's Parametrization of Arthur Packets for p-adic Quasisplit Sp(N) and SO(N)

AbstractWe give a survey on Moeglin's construction of representations in the Arthur packets for p-adic quasisplit symplectic and orthogonal groups.The emphasis is on comparing Moeglin's parametrization of elements in the Arthur packets with that of Arthur.

On Fourier coefficients of certain residual representations of symplectic groups

In the theory of automorphic descents developed by Ginzburg, Rallis and Soudry in [GRS11], the structure of Fourier coefficients of the residual representations of certain special Eisenstein series plays important roles. Started from [JLZ13], the authors are looking for more general residual representations, which may yield more general theory of automorphic descents. In this paper, we investigate the structure of Fourier coefficients of certain residual representations of symplectic groups, corresponding to certain interesting families of global Arthur parameters. On one hand, the results partially confirm a conjecture proposed by the first named author in [J14] on relations between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. On the other hand, the results of this paper can be regarded as a first step towards more general automorphic descents for symplectic groups, which will be considered in our future work.

The non-tempered [formula omitted] Arthur parameter and Gross–Prasad conjectures

We provide a construction of local and automorphic non-tempered Arthur packets AΨ of the group SO(3,2) and its inner form SO(4,1) associated with Arthur's parameterΨ:LF×SL2(C)→O2(C)×SL2(C)→Sp4(C) and prove a multiplicity formula. We further study the restriction of the representations in AΨ to the subgroup SO(3,1). In particular, we discover that the local Gross–Prasad conjecture, formulated for generic L-packets, does not generalize naively to a non-generic A-packet. We also study the non-vanishing of the automorphic SO(3,1)-period on the group SO(4,1)×SO(3,1) and SO(3,2)×SO(3,1) for the representations above. The main tool is the local and global theta correspondence for unitary quaternionic similitude dual pairs.

Some consequences of Arthur's conjectures for special orthogonal even groups

In this paper we construct explicitly through Eisenstein series, a square integrable residual automorphic representation of the special orthogonal group SO2n. We show that this representation comes from an elliptic Arthur parameter and it appears in the space L2(SO2n()\SO2n()) with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger in absolute value than those of the parameter constructed before. The main result is that these parameters cannot be cuspidal. We establish bounds for the eigenvalues of Hecke operators, as consequences of Arthur's conjectures for SO2n. Next, we calculate the character and the twisted characters for the representations that we considered and we prove an identity of traces. Finally, we find the composition of the global and local Arthur's packets associated to our parameter . All the results in this paper are true if we replace by any number field F.