Tuesday, November 7 |
07:00 - 09:00 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort (Dining Hall - Yuxianghu Hotel(御湘湖酒店餐厅)) |
09:30 - 10:30 |
Nicolas Arancibia Robert: Real A-packets from a sheaf theoretic perspective ↓ 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. (Lecture Hall - Academic island(定山院士岛报告厅)) |
10:30 - 11:00 |
Coffee Break (Lecture Hall - Academic island(定山院士岛报告厅)) |
11:00 - 12:00 |
Mishty Ray: Vogan's conjecture for p-adic GL_{n} ↓ 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 GLn. In this talk, we will explore the geometry of the moduli space of Langlands parameters for GLn. We will provide comments on the proof of Vogan’s conjecture in this setting and discuss the scope of further generalizations. (Zoom (Online)) |
12:00 - 13:30 |
Lunch (Dining Hall - Academic island(定山院士岛餐厅)) |
13:45 - 14:45 |
Jessica Fintzen: Representations of p-adic groups and Hecke algebras ↓ 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). (Lecture Hall - Academic island(定山院士岛报告厅)) |
14:45 - 15:00 |
Coffee Break (soft drink only) (Lecture Hall - Academic island(定山院士岛报告厅)) |
15:00 - 16:00 |
Masao Oi: On comparison of Kaletha's and Arthur's toral supercuspidal L-packets of classical groups ↓ 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. (Lecture Hall - Academic island(定山院士岛报告厅)) |
16:00 - 16:30 |
Coffee Break (Lecture Hall - Academic island(定山院士岛报告厅)) |
16:30 - 17:30 |
Emile Okada: Weakly spherical representations and the weak Arthur packet conjecture ↓ Let G be a split reductive p-adic group and 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 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. (Lecture Hall - Academic island(定山院士岛报告厅)) |
18:30 - 20:30 |
Dinner ↓ Outdoor banquet (Academic island(定山院士岛)) |