Abstracts & Program

Title: The Crux of the Biscuit

Abstract: The Atlas software computes the unitary dual of a real reductive group. I will discuss some of the essential points of the Atlas algorithm.  These include: the definition of the c-form; representations of extended groups; orientation numbers; twisted Kazhdan-Lusztig-Vogan polynomials; and the Hodge filtration.

Title: Computing Micro-Packets via the Weyl Group Equivariance of the Characteristic Cycle Map

Abstract: For a real reductive group G, Adams, Barbasch, and Vogan introduce in their book The Langlands Classification and Irreducible Characters for Real Reductive Groups the notion of micro-packets associated to an L-parameter. The definition of micro-packets relies on characteristic cycles, for which no general computational method is currently known. This leaves a general micro-packet without an explicit description.

We show how the equivariance of the characteristic cycle map with respect to a Weyl group action can be exploited to explicitly compute the characteristic cycles, or at least extract partial information about them. Concretely, the talk is organized in two parts. In the first part, we explain how this equivariance can be used to compute the micro-packets containing generic representations. In the second part, we apply these methods to determine all micro-packets for the exceptional group G₂.

Title: Unitary representations of good parity of Sp₂ₙ(F)

Abstract: An irreducible smooth representation of a p-adic group is called unitary if it admits a positive definite invariant hermitian form. 

Classification of such representations is a central problem, called the unitary dual problem, in the representation theory. 

On the other hand, Arthur introduced the notion of representations of Arthur type to classify the local components of the discrete automorphic spectrum. 

In this talk, we will explain that all irreducible unitary representations of good parity of Sp₂ₙ(F), where F is a p-adic field, are of Arthur type. 

This is a joint work with Alberto Minguez in University of Sevilla. 

Title: The local Langlands correspondence through twisted graded Hecke algebras

Abstract: Graded Hecke algebras, which were introduced by Lusztig in order to study the category of unipotent representations of a p-adic group, are graded analogues of affine Hecke algebras.

Twisted graded Hecke algebras are slightly more general versions in which the group algebra of the spherical Weyl group W is replaced by a twisted group algebra of an extension of W by a finite abelian group.

I will explain how twisted graded Hecke algebras naturally occur in both the general representation theory of p-adic groups and the spectral side of the local Langlands correspondence, and can be used to construct the later in a vaste range of situations.

Title: TBA

Abstract: TBA

Title: On irreducible tempered representations of classical p-adic groups

Abstract: In this talk we will present a work in progress on the results concerning the classification of irreducible tempered representations of symplectic or special orthogonal groups over non-archimedean local fields of characteristic zero. Arthur's classification of irreducible tempered representations was followed by the work of Moeglin, Xu and Atobe which made the constituents of Arhur packets explicit. On the other hand, Tadić's and Moeglin's classification of irreducible square integrable representations was followed by the work of Jantzen who extended the classification to irreducible tempered representations in terms of admissible quadruples. 

To describe our results in more detail, let π(φ,ε) denote an irreducible tempered representation parametrized by an L-parameter φ and an epsilon function ε. Our main goal is to give results which characterize the values of the epsilon function ε based on the distinguished constituent of its Jacquet module. These constituents are described in Jantzen's work on tempered representations, which we transfer to Arthur's theory by showing that the epsilon function ε restricts to the epsilon function from the admissible quadruple of Jantzen. Moreover, we simplify obtained characterizations based on the work of Atobe on the semi-simplification of the Jacquet modules of irreducible tempered representations. This is joint work with Igor Ciganović and Marko Tadić.

Title: Unitary dual problem for p-adic classical groups via theta lifts

Abstract: Classifying irreducible unitary representations is one of the most important problems in representation theory of reductive groups. Recently, Atobe--Minguez showed that for symplectic groups and split odd orthogonal groups, an irreducible good parity representation is unitarizable if and only if it is of Arthur type. We generalize their beautiful theorem to other classical groups (possible non-quasi-split) using theta lifts. This is a joint work with Jialiang Zou.

Title: Triality and Adjoint Lifting for GL(3)

Abstract: We show the existence of  the adjoint lifting of cuspidal automorphic representations from GL(3) to GL(8) as a consequence of the theory of twisted endoscopy, applied to the triality automorphism of the adjoint group of type D4. 

We also describe the possible isobaric decomposition of the adjoint lifts. Finally, we discuss applications to the strong Artin conjecture for 3-dimensional tetrahedral Galois representaitons and to the Ramanujan bounds for GL(3). 

We apologize for speaking about 3 at a conference devoted to 1 & 2. 

Title: On the (ir)reducibility of the big theta lift

Abstract: Let F be a non-archimedean local field of characteristic zero. We study theta correspondence for (complex) representations of symplectic–even orthogonal dual reductive pairs over F ; more specifically, the big theta lifts. The explicit description of the small theta lifts, in this situation, is completely known (in terms of their Langlands data), but there are some information about the theta correspondence which are still largely unknown or are currently studied (e.g. the behaviour with respect to various models or periods), the most obvious one being first the question of the (ir)reducibility the big theta lift and then its complete description. We address the state of the affairs concerning the knowledge about big theta lifts (and why we care): from some old results, to some new contributions, e.g. of Chen and Zou, and of the author. If time permits, we tackle an analogous question for certain instances of the exceptional theta correspondences.

Title: Unitary dual of GLₙ(F), F non-archimedean local field -- a look at yesteryear

Title: On Arthur representations and the unitary dual problem

Abstract: In this talk, I will introduce a new conjecture describing the structure of the unitary dual in terms of Arthur representations for connected reductive algebraic groups defined over any non-Archimedean local field of characteristic zero. This conjecture provides a candidate set for the unitary dual, constructed from Arthur representations. For classical groups, we develop an explicit algorithm to generate this candidate set. Evidence for its exhaustiveness includes compatibility with the known generic unitary dual, unramified unitary dual, and low-corank representations. As further support, we verify the conjecture for the unitary dual of the exceptional group of type G2. This is a joint work with Alexander Hazeltine, Dihua Jiang, Chi-Heng Lo, and Qing Zhang. 

Title: The FPP conjecture for p-adic groups

Abstract: The FPP conjecture, proposed by J. Adams, S. Miller, and D. Vogan and proved by D. Davis and L. Mason-Brown, imposes a strong upper bound on the infinitesimal characters of unitary representations of real reductive groups. In this talk, I will introduce a p-adic analogue of the FPP conjecture and sketch a proof under the assumption that the conjectural local Langlands correspondence satisfies the Kazhdan–Lusztig hypothesis. This is joint work with Dihua Jiang, Baiying Liu, and Lucas Mason-Brown.

Title: The Unitarity of Arthur Packets for Real Reductive Groups

Abstract: In the 1980s, Arthur conjectured the existence of some sets (now "Arthur packets") of irreducible representations of a real reductive group satisfying an array of remarkable properties. These sets were defined in the 1990s by Adams, Barbasch, and Vogan. A longstanding conjecture, originally due to Arthur, is that all Arthur packets consist of unitary representations. In this talk, I will sketch a proof of this conjecture. The main new idea is a "Jordan decomposition" for Arthur packets: a canonical two-step process for realizing the members of an Arthur packet via real parabolic and cohomological induction from a unipotent Arthur packet for a (twisted) Levi subgroup. This reduces the question of unitarity to the case of unipotent Arthur packets, where unitarity is already known (by  Adams-Arancibia-Mezo, Adams-van Leeuwen-Miller-Vogan, Arthur, Barbasch, Barbasch-Ma-Sun-Zhu, and Davis-Mason-Brown). This is joint work with Jeff Adams, Andrei Ionov, and David Vogan.

Title: Weak Arthur Packets for classical p-adic groups

Abstract: Weak Arthur packets consist of the minimal representations at certain special infinitesimal characters and, in the real setting, account for many of the unitary representations that one might call ‘unipotent’: the building blocks of the unitary dual. In this talk, I will describe joint work with Maxim Gurevich investigating weak Arthur packets for the split groups Sp(2n) and SO(2n+1) in the p-adic setting. We show that each weak Arthur packet is a union of endoscopic Arthur packets whose SL(2) parameters describe orbits in ‘special’ relative position, and is the 'A-packet closure' of the co-tempered representations that admit fixed vectors under a (not necessarily special) maximal compact open subgroup. Along the way we will see that representations admitting such fixed vectors enjoy several striking properties, and admit a simple description in terms of Lusztig’s canonical quotient.

Title: On the unitarizability of spherical Arthur representations 

(Joint work in progress with V. Heiermann and Marcelo de Martino)

Abstract: Let G be a split simple group over a number field F, with Borel subgroup B=AN. We give a conceptual proof that an appropriate residue of the unramified Borel Eisenstein series at half the middle element c of a standard sl₂ triple corresponding to a distinguished unipotent orbit of the dual group, is square integrable. As a well known application one concludes from this that the spherical Arthur representation of G(Fᵥ) with infinitesimal character cᵥ is unitarizable, at all local places v of F.

Similar results have been established previously by several authors in various special cases (Langlands, Jacquet, Kim, Moeglin, Miller, Green-Miller-Vanhove), and more recently, by a uniform method, by Kazhdan-Okounkov.

We will explain our method, which is quite elementary, and reflect on possible generalisations.

Title: On automorphic duals

Abstract: In our talk we will recall briefly of the role which was played by particular representations in development of a well known Gelfand’s concept of harmonic analysis on locally compact groups. From the very beginning of the work on unitary duals of reductive groups of arbitrary high ranks (like general linear groups), automorphic representations played a very important role. In our talk we will try to give some basic idea of the difference between the automorphic duals in the case of the general linear groups and the classical groups, in particular, regarding the topology.

Title: Unitary representations of (real) reductive groups

Abstract: Suppose G is real reductive. Difficult representations of G appear (by Langlands) in a countable collection of families, each of which is indexed by the elements ν of a (finite-dimensional) real vector space Vⱼ (with j the index for the countable family). The representations pⱼ(ν) all live on a common infinite-dimensional complex vector space Xⱼ. On Xⱼ there is a real-analytic family of hermitian forms <,>(ν). Main points are

1. The representation pⱼ(ν) is unitary iff <,>(ν) is semidefinite.

2. <,>(0) is positive definite.

3. The signature of <,>(ν) can change only on certain hyperplanes in Vⱼ.

The hyperplanes provide a beautiful partition of Vⱼ into faces, each of which is an open simplex. The signature of <,>(ν) is constant on each face.

So understanding unitary representations of G means

A) understanding these simplicial decompositions, and

B) understanding how the signatures vary as one crosses the hyperplanes.

I'll try to explain (A) thoroughly, and to say a little about (B).

Title: Unitary Shimura Correspondence for Real Reductive Groups

Abstract: Let G be a connected real reductive group. Motivated by the work of Shimura and Kazhdan–Patterson, one expects a close relationship between the unitary representations of G and those of G′, where G′ is a nonlinear cover of a real form of the Langlands dual of G. Such correspondence has previously been pursued for various classes of representations, such as the spherical complementary series and trivial representations.

In this talk, we investigate this correspondence for special unipotent representations in two cases: (1) when G is a complex classical group, and (2) when G is a simply laced real group (following the framework of Adams–Herb). We show that these lifts map certain stable sums of special unipotent representations of G to stable sums of genuine unipotent representations of G′. When G is a complex group of Type C and Type D (so that G' is the spin group), we further characterize when the correspondence is non-zero. In such cases, we show it yields the genuine unipotent representations of G' defined by Barbasch and Brega.

This is joint work with K.Y. Chan, W.-Y. Tsai, and H. Zhang.

Title: A correspondence of Arthur packets between real unitary groups and p-adic special odd orthogonal groups

Abstract: We establish an explicit correspondence of certain Arthur packets between real unitary groups and p-adic special odd orthogonal groups. As an application, one can compute the Arthur packets of real unitary groups by translating the results from the p-adic side. The main feature of our proof is to relate the Zuckerman's translation functor on the real side with the Jacquet functor on the p-adic side. To achieve this, we construct a correspondence of stacks of Langlands parameters with fixed infinitesimal characters between the relevant real and p-adic groups. It also allows us to relate the Kazhdan-Lusztig polynomials and the microlocal geometry between real and p-adic cases. This is joint work with Taiwang Deng, Chang Huang and Qixian Zhao.

Title: Special unipotent representations of real classical groups

Abstract: I will give a detailed account of the project with Dan Barbasch, Jia-Jun Ma, and Binyong Sun, in which we classify all special unipotent representations of a real classical group in the sense of Arthur and Adams-Barbasch-Vogan. The classification is achieved via the following two steps: 

1. Counting and parameterizing all such representations combinatorially, by Kazhdan-Lusztig-Vogan theory and the theory of coherent continuation;  

2. Inductively constructing them based on the parameter by a certain ``going down” algorithm of parameters and then “going up” of representations by theta lifting. 

As a direct consequence of the classification, we establish for all real classical groups the Arthur and Adams-Barbasch-Vogan conjecture for unipotent A-parameters, namely all special unipotent representations of real classical groups are unitarizable. The unitarity of special unipotent representations is independently due to Adams-Arancibia-Mezo (for quasi-split orthogonal and symplectic groups) and Arancibia-Mezo (for quasi-split unitary groups). For these classical groups, they also establish unitarity (of representations in the ABV packets) for all A-parameters.