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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Date/Time : 06 Apr 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Nadya Gurevich, Ben Gurion University of the Negev

The classical Schur-Weyl duality, relating representations of the symmetric group S(r) and of sl(n), admits a quantum affine generalization. We explain its relation to the Whittaker functor on the category of Iwahori-spherical genuine representations of a central cover of GL(r) of degree n. 

In particular, we deduce that the space of Whittaker functionals on such representation admits an action of an affine quantum group.

This is a joint work with Fan Gao and Edmund Karasiewicz.

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Date/Time : 04 Apr 2023 16:00
Venue : via Zoom
Speaker : Johannes Flake, MPIM Bonn

Join Zoom Meeting
https://nus-sg.zoom.us/j/86446315086?pwd=amtmalVXUnA0bkRnT0haRmZjeGsrQT09

Meeting ID: 864 4631 5086
Passcode: 280937

I will explain a construction of categories with a tensor product using topological operations involving manifolds, or more precisely, cobordisms. Then I will show how to analyze categories like these using algebraic and representation-theoretic ideas. The categories in question include Deligne’s interpolation categories and Khovanov-Sazdanovic’s cobordism categories. We will discuss their Grothendieck rings and, if time permits, their monoidal centers.

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Date/Time : 30 Mar 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Song Jinfeng, National University of Singapore

Let G be a reductive group over an algebraically closed field with positive characteristic, and N be the unipotent radical of a Borel subgroup. The quantum Frobenius splitting introduced by Lusztig gives an algebraic construction for a Frobenius splitting of the unipotent group N. Another important construction on N is the cluster algebra structure established by Geiss-Leclerc-Schroer. In this talk, I will explain that the algebraic Frobenius splitting on N is compatible with its cluster structure. This is based on an ongoing work.

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