Schedule for: 23w5063 - Arthur packets

A set dinner is served daily between 5:30pm and 8:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A brief introduction with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will explain an argument based on the Kazhdan-Lustig-Vogan algorithm to give sufficient condition for the irreducibility of an parabolically induced representation from an irreducible representation on the Levi factor that proves that this is generically the case (with respect to a twisting by a character). The result in the p-adic case is well-known, and at least two proofs are available in the literature (by F. Sauvageot and J-F. Dat), which seems not to be the case for real groups, although the idea of using the KLV-algorithm to prove irreducibility results has been used before (Matsumoto, Gan-Ichino,...). This result simplifies one step in the construction of Arthur packets for real reductive groups (Moeglin-R) and is also used in a recent work of Offen, Matringe and Yang.

In this talk, I will introduce recent progress on certain problems related to local Arthur packets of classical groups. First, I will introduce a joint work with Freydoon Shahidi towards Jiang's conjecture on the wave front sets of representations in local Arthur packets of classical groups, which is a natural generalization of Shahidi's conjecture, confirming the relation between the structure of wave front sets and the local Arthur parameters. Then, I will introduce a joint work with Alexander Hazeltine and Chi-Heng Lo on the intersection problem of local Arthur packets for symplectic and split odd special orthogonal groups, with applications to the Enhanced Shahidi's conjecture, the closure relation conjecture, and a conjecture of Clozel on unramified representations. This intersection problem also has been worked out independently at the same time by Hiraku Atobe.

Lunch is served daily between 12:00am and 1:30pm in the Xianghu Lake National Tourist Resort

Let $\pi$ be a cuspidal automorphic representation of a classical group or a metaplectic group. We show an exact relation between two invariants associated to $\pi$, one called the lowest occurrence index of $\pi$ which is an invariant arising from theta lifts and the other the location of the maximal positive pole of an Eisenstein series attached to $\pi$. As an application, we use this relation to show that certain global Arthur packets cannot contain cuspidal automorphic representations. This can be used to find a Ramanujan bound that measures the non-temperedness of $\pi$.

Consider a reductive group G and its closed subgroup H. A natural question is to study the restriction of representations of G into H in both global and local aspects. Under the theory of local Langlands correspondences, one may expect more refinery structures on local Arthur packets arising from the above questions from the representation side. In this talk, we first discuss certain strongly tempered spherical varieties whose generic local Arthur packets possess the same epsilon dichotomy properties, such as the local Gan-Gross-Prasad Conjecture. Secondly, we propose a general conjectural local multiplicity formula when higher multiplicities occur in split reductive group cases. Moreover, Ben-Zvi, Sakellaridis, and Venkatesh proposed the relative Langlands duality for spherical varieties. Following their proposal, we also investigate their dual spherical varieties and propose questions on the local multiplicity formula of the dual sides, which involves the non-tempered Arthur packets.

We consider the reductive dual pair $Aut(C) \times F_4$ inside a form of $E_7$ over a p-adic field, where $C$ is a quaternion algebra. The theta correspondence arises by restricting the minimal representation of $E_7$ to the dual pair. By computing the Fourier-Jacobi functor and Jacquet modules of the minimal representation we derive precise information about this theta correspondence. As an application we prove a multiplicity one result for $Spin(9)$-periods of $F_4$. Finally, we discuss how the theta correspondence can be used to construct some local Arthur packets for $F_4$. This is joint work with Gordan Savin.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

In his book "The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups", J.Arthur gives a definition of A(rthur)-packets for quasisplit classical groups using techniques from harmonic analysis. Guided by Arthur's ideas, C.P Mok defines A-packets for quasisplit unitary groups. Recently, C. Moeglin and D. Renard have extended the definition of A-packets to all pure inner forms of real classical and unitary groups. For real groups, an alternative approach to the definition of A- packets has been known since the early 90s. This approach, due to Adams-Barbasch-Vogan, relies on sheaf-theoretic techniques instead of harmonic analysis. It has long been expected that the two constructions agree in the cases in which they are both defined: (pure) real forms of classical and unitary groups. In a joint work with J. Adams and P. Mezo, it is finally verified that these two constructions, defined through two fundamentally different points of view, are the same in the mentioned cases. This places sheaf-theoretic methods on equal footing with analytic methods when considering properties of automorphic forms at archimedean places, offering us new perspectives to attack questions that until now had mostly been studied through a harmonic analysis perspective. In this talk we review the proof of the equality between both constructions and if the time permits we explore some of its consequences and mention some future research perspectives that it opens.

Arthur's description of the constituents of square-integrable automorphic representations gave rise to the theory of Arthur packets, or A-packets. Local A-packets are local analogues of the aforementioned global objects; they are sets of unitary irreducible representations of $p$-adic groups attached to local Arthur parameters. When Arthur initially came up with his theory around 1990, the local meaning of his work was of interest. Adams, Barbasch, and Vogan suggested a beautiful geometric appoach to local A-packets for real groups, which was later refined by Vogan for non-archimedean places in the 90s. Vogan’s geometric perspective on the Langlands correspondence establishes a bijection between equivalence classes of smooth irreducible representations of G(F) and simple equivariant perverse sheaves on a moduli space of Langlands parameters. This gives us the notion of an ABV- packet, which is a set of smooth irreducible representations of G(F), but now attached to any Langlands parameter. As of 2013, Arthur has clarified the local theory for his work, which opens up the opportunity to study the relationship between local A-packets and ABV-packets. Conjecturally, ABV-packets generalize local A-packets; we call this Vogan’s conjecture. In recent work joint with Clifton Cunningham, we prove this conjecture for $p$-adic $GL_n$. In this talk, we will explore the geometry of the moduli space of Langlands parameters for $GL_n$. We will provide comments on the proof of Vogan’s conjecture in this setting and discuss the scope of further generalizations.

We show that one can reduce a lot of problems about the (category of) (smooth, complex) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers might already be known or are easier to achieve. More precisely, the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. Some of these blocks, called depth-zero blocks, correspond essentially to blocks in the category of representations of finite groups of Lie type and are much better understood than arbitrary Bernstein blocks. I will discuss a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara in which we show that general Bernstein blocks are equivalent to the much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras. To put everything into context, I will also recall what is known about the explicit construction of (supercuspidal) representations of p-adic groups (and types).

Recently, Kaletha constructed L-packets for a wide class of supercuspidal representations of tamely ramified connected reductive groups. In this talk, I will explain that Kaletha's L-packets coincide with Arthur's L-packets in the case of toral supercuspidal representations of quasi-split classical groups. The strategy is to show that Kaletha's L-packets satisfy the twisted endoscopic character relation, which is the characterization of Arthur's L-packets, by establishing an explicit formula of twisted characters.

Let $G$ be a split reductive p-adic group and $\mathcal{O}$ be a nilpotent orbit of the Langlands dual group. In joint work with Dan Ciubotaru and Lucas Mason-Brown we conjectured that a certain natural collection of representations associated to $\mathcal{O}$ should be a union of Arthur packets. In this talk I will present a proof of this conjecture for symplectic and odd orthogonal groups (also proved independently by Baiying Liu and Chi-Heng Lo). In the process we see that the so-called weakly spherical representations (representations containing a fixed vector with respect to a maximal compact subgroup) play an important role in this picture. This is joint work with Max Gurevich.

Outdoor banquet

The nontempered Gan-Gross-Prasad program aims to study the quotient branching laws for representations in an Arthur packet. I will discuss a new proof to the case of p-adic general linear groups. More precisely, we give a new formulation (and proof) for branching laws for all irreducible representations, and explain how this formulation can give the case for Arthur type representations. If time permits, I will explain how to deduce some branching laws of classical groups from the p-adic GL case.

Let $T$ be a torus defined over a local field $K$ with an algebraically closed residue field $k$. We then have the connected commutative pro-algebraic group $L^+T$ (defined over $k$) constructed from the connected Neron model of $T$. In this talk I will describe the abelian group of multiplicative local systems (or character sheaves) on $L^+T$ and prove that it is canonically isomorphic to the abelian group of inertial local Langlands parameters for $T$. I will then relate this result to the usual local Langlands correspondence for tori via the sheaf-function correspondence. This is joint work with Saniya Wagh.

I will explain my recent work with Masao Oi on parametrizing the local Langlands correspondence using the Kottwitz set and try to justify the assertion that our construction is "natural". I will discuss some instructive examples as well as our partial progress on a generalized endoscopic character identity.

Aubert, Moussaoui, and Solleveld have formulated a version of the Bernstein decomposition for enhanced L-parameters. Their results are the first step in a general strategy to reduce local Langlands correspondences to the supercuspidal case. In this talk I'll explain how to modify the theory, which uses Arthur's enhancements of L-parameters, to fit into Kaletha's framework of rigid inner forms. Along the way I'll give a more conceptual description of the generalized Springer correspondence for disconnected groups, one which is surely known to experts but has not appeared in the literature. This is joint work with Peter Dillery.

The minimal (Weil) representation of the metaplectic group is realized automorphically by theta functional. We generalize the construction of an automorphic functional on various models of minimal representation of simply laced groups of type D and E, stressing the main term and the boundary terms involved. In certain cases this gives rise to generalized Poisson summation formula. This is joint work with David Kazhdan.

Recently, based the early work of Moeglin, Bakic-Hanzer have proved the Adams conjecture, which described the theta correspondence in terms of A-parameters. In this talk, we propose a refined version of Adams conjectures which also takes care of the labellings inside A-packets, and prove it in the stable range case. Our main tool is a corollary of the local intertwining relation. If time permit we will also talk about some potential applications to the metaplectic group. This is based on a joint work with Jialiang Zou.

ABV-packets are candidate Arthur packets constructed from pure geometric method. In this talk, I will report certain calculation of ABV-packets of the exceptional group G_2 and some general results on ABV-packets which contain generic representations. This talk is based on joint work with Cunningham-Fiori, with Cunningham- Dijols-Fiori, and with Hazeltine-Liu-Lo over the past several years.

Let $G$ and $H$ be real reductive groups. To any L-homomorphism $e : {}^L H \rightarrow {}^L G$ one can associate a map $e_*$ from virtual representations of $H$ to virtual representations of $G$. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which $e_*$ exhibits especially nice behavior is the case when $H$ is an endoscopic group. In this talk, I will introduce a more general class of groups which exhibit similar behavior. I will explain how this generalized version of endoscopic lifting can be used to prove the unitarity of many Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.

Special unipotent representations, also known as weakly unipotent representations, are expected to exhaust the unipotent Arthur packets. In a recent collaboration with Dan Barbasch, Binyong Sun, and Chengbo Zhu, we have classified and proved the unitarity of these representations for real classical groups. In this presentation, I will cover the following topics: the counting formula of unipotent representations the reduction to the good parity case the combinatorial parameterization and iterated theta lifting construction of the weak Arthur packet with good parity.