Date/Time : 19 Jan 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Chenyan Wu, University of Melbourne

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 $\pi$ and a character. As an application, We show that certain global Arthur packets cannot contain cuspidal automorphic representations.

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Date/Time : 12 Jan 2023 10:00
Venue : S17 #04-05 (SR2)
Speaker : Stefan Dawydiak, Max Planck Institute for Mathematics in Bonn (MPIM)

In the 1980s, Kazhdan-Lusztig and independently Ginzburg showed that the affine Hecke algebra $H$ can be realized as the Grothendieck group of the category of coherent sheaves on the Steinberg variety. The correct coherent Hecke category was however found only later, by Bezrukavnikov. A crucial ingredient in this coherent categorification of $H$ is viewing the Steinberg variety as a derived scheme.

On the other hand, Lusztig’s asymptotic Hecke algebra $J$ is a certain based ring receiving a map from $H$, via which its representation theory is closely related to that of $H$. In the 1990s, Xi showed that the best-understood direct summand $J_0$ of $J$ can be realized as the Grothendieck group of coherent sheaves on the square of the flag variety, and Ginzburg gave a K-theoretic definition of the map from $H$ to $J_0$. In this talk we will explain how to upgrade this realization to a monoidal category receiving a natural monoidal functor from the coherent Hecke category. A crucial ingredient is again to consider a derived enhancement, this time of the flag variety.

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Date/Time : 11 Nov 2022 14:00
Venue : via Zoom
Speaker : Liping Li, Hunan Normal University

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https://nus-sg.zoom.us/j/85349169300?pwd=bE9Ed2ZLelNHdVpjSFV5YnEwdlJpUT09

Meeting ID: 853 4916 9300
Passcode: 176749

Abstract: In this talk we describe an approach to classify irreducible discrete representations of infinite linear groups, based on Artin’s theorem connecting discrete representations of topological groups and sheaves of modules over atomic sites as well as some recent development in representation stability theory.

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Date/Time : 04 Nov 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Fang Xinyu, National University of Singapore

Abstract: A conjecture by Cohen and Lenstra (1984) predicts a distribution for the class group of a random quadratic field. This was one of the first motivating examples for the probability theory of random groups that arise in number theory. In this talk we will give a brief overview of the theory, and introduce some major results concerning the moment problem as well as universality. Then we will focus on some universality results for random abelian groups that arise as cokernels of random matrices, possibly with additional structure (such as being symmetric, Hermitian or alternating). We show that a method used to compute a formula for the distribution of cokernels of symmetric matrices also works in the case of Hermitian ones, when they are drawn from additive Haar measure.

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Date/Time : 14 Oct 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Le Nhat Hoang, National University of Singapore

Abstract: The local Gan-Gross-Prasad (GGP) conjecture studies a family of restriction problems for classical groups and proposes precised answers to these problems using the local Langlands correspondence. It also has a twisted variant in the Fourier-Jacobi case. In this talk, I will present my proof for the tempered part of the local twisted GGP conjecture for (GL(V),U(V)), which is obtained by adapting Waldspurger and Plessis’s method to construct a local trace formula and comparing its spectral and geometric sides. Moreover, I will talk about how the geometric multiplicity formula looks like in the remaining case, when we consider an another quadratic extension K/F and have the local twisted GGP conjecture for (U(V)(K),U(V)).

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Date/Time : 07 Oct 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Chung-Ru Lee, National University of Singapore

Abstract: One of the main application of the Relative Langlands Program is providing a description of distinguished representation, which itself has profuse interpretations. In this talk series I will introduce a method to study such problem through the comparison of relative trace formulae, where the theory of endoscopy plays a significant role. In particular, the study of representations on unitary groups that are distinguished by an orthogonal subgroup remains mysterious due to the presence of a type N-spherical root, which requires extra caution when dealt with.

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Date/Time : 30 Sep 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Chung-Ru Lee, National University of Singapore

Abstract: One of the main application of the Relative Langlands Program is providing a description of distinguished representation, which itself has profuse interpretations. In this talk series I will introduce a method to study such problem through the comparison of relative trace formulae, where the theory of endoscopy plays a significant role. In particular, the study of representations on unitary groups that are distinguished by an orthogonal subgroup remains mysterious due to the presence of a type N-spherical root, which requires extra caution when dealt with.

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Date/Time : 16 Sep 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Emile Takahiro Okada, National University of Singapore

Abstract: In my talk last year I introduced new methods for computing the wavefront set of depth 0 representations using local wavefront sets. In this series of talks I will begin by briefly recalling the main results from that talk before I go on to outline new results and open questions that have come from these ideas. These will fall into roughly 3 categories: the structure theory of nilpotent orbits, new results on the wavefront set, and finally applications to Arthurs packets.

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Date/Time : 09 Sep 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Emile Takahiro Okada, National University of Singapore

Abstract: In my talk last year I introduced new methods for computing the wavefront set of depth 0 representations using local wavefront sets. In this series of talks I will begin by briefly recalling the main results from that talk before I go on to outline new results and open questions that have come from these ideas. These will fall into roughly 3 categories: the structure theory of nilpotent orbits, new results on the wavefront set, and finally applications to Arthurs packets.

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