Date/Time : 19 Apr 2022 15:00
Venue : via Zoom
Speaker : Xiaoyu Zhang, University of Duisburg-Essen.

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https://nus-sg.zoom.us/j/88995853527?pwd=MHZNbFVGTy93QVJEMGFtdEtLdGJGdz09

Meeting ID: 889 9585 3527
Passcode: 388541

Abstract: Theta correspondence is a powerful tool in the theory of automorphic representations. Many fundamental problems in the complex coefficient case have been resolved thanks to the work of many people and have led to very deep applications in Langlands program and other problems in number theory. In this talk, we consider theta correspondence of orthogonal-symplectic type. Given the integral structures of the spaces of automorphic forms, we will show that if an automorphic form on orthogonal group is non-zero mod p, so is its theta lift. The proof involves a comparison of the Bessel period of the theta lift and the toric period of the original automorphic form and is reduced to an equidistribution result of CM points on the orthogonal group. If time permits, we will also discuss some problems related to these results.

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Date/Time : 12 Apr 2022 14:00
Venue : via Zoom
Speaker : Yu Jun, Peking University

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https://nus-sg.zoom.us/j/89178679746?pwd=NUNjUVdTcWhLZ082di9tM0I1cDRXQT09

Meeting ID: 891 7867 9746
Passcode: 548511

Abstract: Given a quasisplit reductive linear algebraic group $\mathbb{G}$ over a number field $F$, this talk concerns the determination of the functional source group $H_{\pi}$ of an automorphic representation $\pi$ of Ramanujan type of $\mathbb{G}(\mathbb{A}_{F})$. Assuming that $\mathbb{G}$ is split, a unitary group or an orthogonal group and $\pi$ neither comes from the Eisenstein series nor comes from endoscopy, we show that $H_{\pi}$ is determined up to element-conjugacy by its dimension datum alone. We also show results concerning linear independent of dimension data of such $H_{\pi}$.

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Date/Time : 07 Apr 2022 14:00
Venue : S17 #04-05 (SR2)
Speaker : Weiqiang Wang, University of Virginia

Abstract: Symmetric pairs over a local field admit a quantum deformation known as quantum symmetric pairs of affine type, (U, U^i). Here U is a Drinfeld-Jimbo affine quantum group (a quantum deformation of a loop algebra), and it admits a Drinfeld (loop) presentation. In this talk, we give a Drinfeld presentation for the affine i-quantum group U^i. Along the way, we shall construct various root vectors in U^i and describe their classical limits. This is based on joint work with Ming Lu (Sichuan) and Weinan Zhang (Virginia).

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Date/Time : 05 Apr 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Huanchen Bao, National university of Singapore

Abstract:  The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.

We show that the J-totally nonnegative flag variety has a cellular decomposition into totally positive J-Richardson varieties. Moreover, each totally positive J-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.

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Date/Time : 31 Mar 2022 09:00
Venue : via Zoom
Speaker : Stefan Dawydiak, University of Toronto

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https://nus-sg.zoom.us/j/89752886605?pwd=Z1V3N2lPbEtodjViZXVmMXRCNGFpZz09

Meeting ID: 897 5288 6605
Passcode: 343140

Abstract:  In the early 1980s Lusztig defined a ring J that serves as a limit of an affine Hecke algebra as the parameter v tends to 0, and showed that J can be realized as a subalgebra of a completion of the affine Hecke algebra, with elements of J being infinite linear combinations of elements of the affine Hecke algebra with Laurent series coefficients. Using work of Braverman-Kazhdan and the Plancherel formula for p-adic groups, we prove that the above coefficients are in fact rational functions whose denominators all divide a fixed polynomial that depends only on the affine Weyl group. We conjecture, and prove in the case that the p-adic group is GL_n, that this polynomial is precisely the Poincare polynomial of the finite Weyl group of G. We will explain a possible application to the representation theory of affine Hecke algebras at root of unity.

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Date/Time : 22 Mar 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jiajun Ma, Xiamen University

Abstract: The program of classification of primitive ideals in the universal enveloping algebra was completed by Barbasch-Vogan, Duflo, Lusztig, and Joseph in the 1980s. A representation of a real reductive group is called special unipotent if its annihilator idea is the maximal primitive ideal attached to a nilpotent orbit in its Langlands dual.

This talk will explain how to count the number of unipotent representations using coherent continuation representations. In the classical group case, we will relate the counting problem with painted partition and bipartition.

This is joint work with Dan Barbasch, Binyong Sun, and Chengbo Zhu.

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Date/Time : 15 Mar 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jiajun Ma, Xiamen University

Abstract: Consider theta correspondence of type I dual pair over a finite field F_q (q odd). Aubert-Michel-Rouquier deduced the explicit theta correspondence between unipotent representations for unitary groups using character theory and proposed a conjecture for the symplectic group-even orthogonal group pair. Following AMR’s approach, Shu-Yen Pan recently proved the conjecture using the combinatorics of symbols.

This talk will give a conceptually simpler approach to the question: directly compute the natural Hecke algebra module structure arising in the theta correspondence. We will see that the normalization of Hecke algebra is related to the first occurrence index, which leads to yet another proof of the conservation relation.

This is joint work with Jialiang Zhou and Congling Qiu.

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Date/Time : 08 Mar 2022 09:00
Venue : via Zoom
Speaker : Weinan Zhang, University of Virginia

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https://nus-sg.zoom.us/j/84683629566?pwd=SXBHQzdmcUoxLzhrQ3pqU29ERS9KZz09

Meeting ID: 846 8362 9566
Passcode: 746935

Abstract: Introduced by Lusztig in the early 1990s, the braid group symmetries constitute an essential part in the theory of quantum groups. The i-quantum groups are coideal subalgebras of quantum groups arising from quantum symmetric pairs, which are now viewed as natural generalizations of quantum groups. In this talk, I will present our construction of relative braid group symmetries (associated to the relative Weyl group of a symmetric pair) on i-quantum groups of arbitrary finite types. These new symmetries inherit most properties of Lusztig’s symmetries, including compatible relative braid group actions on modules. This is joint work with Weiqiang Wang.

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Date/Time : 22 Feb 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : David Hansen, Max Planck Institute for Mathematics in Bonn

Abstract:  I’ll talk about some recent foundational developments in etale cohomology:

i) A flexible six-functor formalism for “Zariski-constructible” sheaves on rigid spaces (joint work with Bhargav Bhatt).

ii) A new “relative” variant of perverse sheaves (joint work with Peter Scholze).

If time permits, I’ll mention some tantalizing open problems in these directions.

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