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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Date/Time : 08 Feb 2022 14:00
Venue : via Zoom
Speaker : Wang Zhicheng, Soochow University

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https://nus-sg.zoom.us/j/83614500983?pwd=U1psUGRnbVRiN0ZyZ2pzWU1aWjNDQT09

Meeting ID: 836 1450 0983
Passcode: 893893

Abstract: In this talk, we will discuss the Gan-Gross-Prasad problem for finite classical groups. Our results provide complete answers for unipotent representations, and we obtain the explicit branching laws for these representations. Moreover, for arbitrary representations, we give a formula to reduce the Gan-Gross-Prasad problem to the restriction problem of Deligne-Lusztig characters.

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Date/Time : 25 Jan 2022 15:00
Venue : via Zoom
Speaker : Haiping Yang, Imperial College London

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https://nus-sg.zoom.us/j/82929931113?pwd=dFlvb3U4SU9ta05jdCtPdWQrOVFodz09

Meeting ID: 829 2993 1113
Passcode: 528816

Abstract: The category of D-mods was introduced in the 1970s and has been a huge area of interest. While it is interesting on its own right, it also has many applications in representation theory and geometry. In particular, holonomicity can be used to show finiteness of certain invariants. For example, finiteness of Skein modules of 3-manifolds [GJS] and finiteness of Poisson (co)homology of certain Poisson algebras [ES, PS].

In the first part of this talk, we are going to introduce the category of D-modules on a (not necessarily smooth) variety. In particular, we will see the derived category is compactly generated, and therefore construct a derived equivalence to the category of dg-modules over an explicit dg-ring. When the variety has only cuspidal singularities, we show that the equivalence is the derived version of the abelian equivalence described by Ben-Zvi and Nevins.

In the second part of the talk, we are going to explore the holonomicity idea and define Hochschild-de Rham homology. This version of Hochschild homology behaves better for a symplectic singularity. In particular, we are going to see a version of Nest-Tsygen theorem for symplectic singularities.

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Date/Time : 18 Jan 2022 09:00
Venue : via Zoom
Speaker : Gaston Naredo, University of Sydney

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https://nus-sg.zoom.us/j/81663160111?pwd=WnVRdlh0SVV4Z1M1L24rT3F2OXArUT09

Meeting ID: 816 6316 0111
Passcode: 390667

Abstract: In this talk, I will explore the top-heaviness and unimodality behaviors of certain sequences. I will explain (following the work of Björner and Ekedahl) why Deligne’s hard Lefschetz theorem implies the top-heaviness theorem for the Betti numbers of a proper variety X having an algebraic cell decomposition.

The unimodality conjecture is false even for “type A” Schubert varieties. However, I believe the conjecture is true for dominant alcoves in affine Weyl groups. I will show examples of this in ranks 2, 3, and 4. I will show some geometric visualisations in ranks 2 and 3 and explain the research directions I am taking to address this problem.

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Date/Time : 11 Jan 2022 15:00
Venue : via Zoom
Speaker : Lee Si Ying, Harvard University

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https://nus-sg.zoom.us/j/88612905478?pwd=cjlrSTRiT3MvYmE5WmlNNGJKdERHQT09

Meeting ID: 886 1290 5478
Passcode: 587672

Abstract: The well-known classical Eichler-Shimura relation for modular curves asserts that the Hecke operator $T_p$ is equal, as an algebraic correspondence over the special fiber, to the sum of Frobenius and Verschiebung. Blasius and Rogawski proposed a generalization of this result for Shimura varieties with good reduction at $p$, and conjectured that the Frobenius satisfies a certain Hecke polynomial. I will talk about a recent proof of this conjecture for a large class of Shimura varieties of abelian type, and how this proves semisimplicity of cohomology for some Shimura varieties.

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Date/Time : 24 Nov 2021 15:00
Venue : via Zoom
Speaker : Jonathan Gruber, École Polytechnique Fédérale de Lausanne

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https://nus-sg.zoom.us/j/85892358313?pwd=L2ZESjRtQ0xyV244T2RkZEt4UWdXUT09

Meeting ID: 858 9235 8313
Passcode: 873255

Abstract: Let U be a quantum group at a primitive l-th root of unity, obtained by extension of scalars from Lusztig’s integral form of the quantized enveloping algebra of a complex simple Lie algebra. A key role in the representation theory of U is played by the so-called tilting modules. The category Tilt(U) of tilting U-modules is closed under tensor products, and in many respects, its monoidal structure is better understood than that of the category Rep(U) of all finite-dimensional U-modules.

In this talk, I will first explain how a U-module can be represented by a unique minimal complex of tilting modules. Then I will discuss how minimal complexes can be used to transfer information about the structure of tensor products from Tilt(U) to Rep(U).

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