Date/Time : 06 Oct 2023 16:30
Venue : S17 #05-11 (SR5)
Speaker : Bryan Wang Peng Jun, National University of Singapore

Recent work of Ben-Zvi, Sakellaridis and Venkatesh (BZSV) predicts the existence of a duality in the relative Langlands program, indexed by a class of Hamiltonian spaces with reductive group action. I will first give a rough introduction to the BZSV-duality, focusing on motivations and a series of motivating examples. Then, I will explain a local incarnation of the BZSV-duality in the setting of branching problems, and how the generalised Whittaker (or Gelfand-Graev) models serve as the prototypical example. Finally, I will present a new family of examples of BZSV-duality, which can so far be verified in this local setting. Time permitting, I will explain the proof, which proceeds via results of Gomez and Zhu, and an interesting interpretation in the framework of Hamiltonian spaces. This is based on my thesis work supervised by Wee Teck Gan.

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Date/Time : 13 Sep 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Shuichiro Takeda, Osaka University

Let X=H\G be a symmetric variety over a p-adic field, where we assume G is split. In this talk, I will introduce a complex group G_X^\vee, which naturally gives rise to a map from G_X^\vee\time SL_2 to the Langlands dual group G^\vee. Further, I will talk about how this map is related to H-distinguished representations of G through numerous examples, especially in relation to the theory of Kato-Takano on H-relative matrix coefficients.

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Date/Time : 13 Sep 2023 16:30
Venue : S17 #04-06 (SR1)
Speaker : Arghya Sadhukhan, National University of Singapore

First introduced by Rapoport, affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties in the context of an affine root system. Understanding their geometry has been crucial in studying reduction of Shimura varieties and have found applications in the Langlands program; for example, description of their irreducible components has been employed to understand cycles on the special fiber a la Xiao-Zhu, whereas knowledge about their connected component has been useful in Kisin’s proof of the Langlands-Rapoport conjecture (in the hyperspecial level). In this talk, we will discuss a dimension formula for certain union of affine Deligne-Lusztig varieties of parahoric level in the case of a quasi-split group, as well as some partial results in the non-quasi split case, obtained from utilizing certain combinatorial tools arising from quantum Schubert calculus.

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Date/Time : 08 Sep 2023 16:30
Venue : S17 #04-06 (SR1)
Speaker : Zhu Chengbo, National University of Singapore

I will explain how to count irreducible representations with a given infinitesimal character and a given bound of the complex associated variety. The main conceptual tool for the counting is coherent families of representations. For the category of Harish-Chandra modules, the concept was developed by Schmid, Zuckerman, Speh-Vogan, and Vogan. Putting together all coherent families (based on a translate of the weight lattice), one obtains a representation of a certain integral Weyl group. The key to the counting lies in understanding cell structures of this representation (called Harish-Chandra cells), and this is where Kazhdan-Lusztig theory, more specifically the theory of primitive ideals (as developed by Joseph and Barbasch-Vogan), double cells and special representations (in the sense of Lusztig), comes into play. The end result is a counting inequality on irreducible representations, which is shown to be an equality in important cases (e.g. all classical groups). This is joint work with Dan Barbasch, Jia-Jun Ma and Binyong Sun, and may be considered as Part I of our project to construct and classify special unipotent representations of all classical groups (with unitarity as a direct consequence).

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Date/Time : 08 Sep 2023 14:45
Venue : S17 #05-11 (SR5)
Speaker : Atsushi Ichino, Kyoto University

The Bertolini-Darmon-Prasanna formula is a p-adic analog of the Gross-Zagier formula, and relates the special value of the p-adic L-function with the p-adic Abel-Jacobi image of some cycle. In this formula, the local root number at infinity is assumed to be -1. In this talk, we discuss the case when this sign is +1. This is joint work in progress with Kartik Prasanna.

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Date/Time : 25 Aug 2023 14:00
Venue : S17 #04-04 (SR3)
Speaker : Zhang Lei, National University of Singapore

In this talk, I will report our progress on the joint project with Chen Wan (Rutgers University—Newark). In this project, we investigate a family of strongly tempered spherical varieties and conjecture that they share analogous global and local properties with well-known GGP models. Furthermore, over local fields, we give a general epsilon dichotomy conjecture for strongly tempered spherical subgroups without type N root.

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Date/Time : 03 Aug 2023 14:30
Venue : S17 #04-05 (SR2)
Speaker : Dinesh Thakur, University of Rochester

Without assuming any background, we will explain how some basic techniques and results from computer science help us in understanding the nature of some naturally occurring numbers.

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Date/Time : 03 Aug 2023 16:00
Venue : S17 #04-05 (SR2)
Speaker : Wong Tian An, University of Michigan, Deerfield

Following Langlands’ original Beyond Endoscopy proposal, Arthur has outlined two refinements of the stable trace formula: the r-trace formula and the primitive trace formula. Spectrally, the first should detect specific functorial transfers from smaller groups, while the second should consist of certain primitive automorphic forms. In this talk, I will give a friendly introduction to these ideas. Then I will present a framework for establishing each of them separately, and report on some progress along these lines.

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Date/Time : 13 Apr 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Lucas Mason-Brown, University of Oxford

Let G and H be real reductive groups. To any L-homomorphism e: H^L \to G^L 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 L-homomorphisms which exhibit similar behavior to the endoscopic case. I will explain how this more general notion of endoscopic lifting relates to the theory of cohomological induction. I will also explain how this generalized notion 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.

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