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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Date/Time : 23 Nov 2021 09:00
Venue : via Zoom
Speaker : Rahul Singh, University of Pittsburgh

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https://nus-sg.zoom.us/j/84886649094?pwd=Mi9uUE1uREd5eVI3RG55c0JHZmlMQT09

Meeting ID: 848 8664 9094
Passcode: 337125

Abstract: A Higgs bundle over an algebraic curve is a vector bundle with a twisted endomorphism. An important question is to calculate the volume of the groupoid of Higgs bundles over finite fields. In 2014, Olivier Schiffmann succeeded in finding the corresponding generating function and together with Mozvogoy reduced the problem to counting pairs of a vector bundle and a nilpotent endomorphism. It was generalized recently by Anton Mellit to the case of Higgs bundles with regular singularities. An important step in Mellit’s calculations is the case of P^1 and 2 marked points, which allows him to relate the corresponding generating function with the Macdonald polynomials

It is a natural question to generalize Mellit’s calculations to arbitrary reductive groups. In my work, I consider the case of P^1 with 2 points for an arbitrary split connected reductive group G over F_q. Firstly, I give an explicit formula for the number of F_q-rational points of generalized Steinberg varieties of G. Secondly, for each principal G-bundle over P^1, I give an explicit formula counting the number of triples consisting of parabolic structures 0 and ∞ and compatible nilpotent sections of the associated adjoint bundle.

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Date/Time : 18 Nov 2021 09:00
Venue : via Zoom
Speaker : Yusheng Lei, Boston College

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https://nus-sg.zoom.us/j/83129514590?pwd=U2VWNnhCUGtQR1NpUGo5Zlh3TUhxdz09

Meeting ID: 831 2951 4590
Passcode: 415969

Abstract:  In 2003, Bump-Friedberg-Ginzburg constructed the generalized global theta representation on a metaplectic double cover of an odd special orthogonal group. which was used later to construct the non-minimal theta liftings between double covers of orthogonal groups. This can be viewed as a generalization of the classical theta correspondence. In particular, it enjoys the tower property similar to the Rallis tower in the classical setting. This raises the question of when the first non-zero lifting will occur for a fixed theta tower. Bump-Friedberg-Ginzburg analyzed this problem when the automorphic representations are generic. In this talk, we will show the way to construct such theta liftings and talk about some progress towards understanding the non-generic cases.

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Date/Time : 11 Nov 2021 09:00
Venue : via Zoom
Speaker : Yaping Yang, University of Melbourne

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https://nus-sg.zoom.us/j/85407853911?pwd=b3E4ZXNERkFaOGtQalNFKzN4MldrUT09

Meeting ID: 854 0785 3911
Passcode: 295101

Abstract:  In this talk, I will explain a connection between stable homotopy theory and representation theory. I will focus on one application of this idea to a problem arising from the modular representation theory. More explicitly, we study a family of new quantum groups labelled by a prime number and a positive integer constructed using the Morava E-theories. Those quantum groups are related to Lusztig’s 2015 reformulation of his conjecture from 1979 on character formulas for algebraic groups over a field of positive characteristic. This talk is based on my joint work with Gufang Zhao.

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Date/Time : 27 Oct 2021 15:00
Venue : via Zoom
Speaker : Tomasz Przezdziecki, University of Edinburgh

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https://nus-sg.zoom.us/j/81363430790?pwd=ODk2Qko1aGhPVW9LS0Y0aDJrTG9uZz09

Meeting ID: 813 6343 0790
Passcode: 720832

Abstract: We consider a family of KLR-type algebras associated to the infinite linear quiver with an involution and a framing. We approach their representation theory through categorification and combinatorics. Orientifold KLR algebras are known to categorify highest weight modules over the Enomoto-Kashiwara algebra and their canonical bases. Using a new shuffle realization of these modules, together with a combinatorial construction of their PBW and canonical bases in terms of Lyndon words, we are able to classify irreducibles and prove finite global dimension.

The second part of the talk is devoted to the connection between orientifold KLR algebras and quantum Kac-Moody symmetric pairs. The key role is played by a Schur-Weyl duality type functor, generalizing a construction due to Kang, Kashiwara and Kim. We explain the connection between the framing in the definition of the orientifold KLR algebra, parameters of a coideal subalgebra, and the poles of meromorphic K-matrices. We present some properties of this functor and formulate a conjecture that, under appropriate assumptions, it induces an equivalence of categories.

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Date/Time : 20 Oct 2021 14:00
Venue : via Zoom
Speaker : Wen-Wei Li, Peking University

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https://nus-sg.zoom.us/j/81065287062?pwd=ajhROWc5MWV1RVQzQ1ZkS2NNcHp1Zz09

Meeting ID: 810 6528 7062
Passcode: 121544

Abstract:  A theory of endoscopy for the metaplectic covering Mp(2n) of Sp(2n) was proposed by the author ten years ago, and the elliptic part of the Arthur-Selberg trace formula has been stabilized since then. In this talk, I will give an overview of the stabilization of the full trace formula (arXiv:2109.06581) for these coverings, which is indispensable for global applications. This is largely inspired by the prior works of Arthur and Moeglin-Waldspurger for linear reductive groups.

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Date/Time : 13 Oct 2021 14:00
Venue : via Zoom
Speaker : Hankyung Ko, Uppsala University

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https://nus-sg.zoom.us/j/86403704548?pwd=ZWFSVHpqMVd0TkNNVTM0ZEg0MzhSdz09

Meeting ID: 864 0370 4548
Passcode: 234932

Abstract:  A Coxeter system is a presentation of a group by generators and a specific form of relations, namely the braid relations and the reflection relations. The Coxeter presentation leads to, among others, a similar presentation of the (Iwahori-)Hecke algebras and the Kazhdan-Lusztig theory, which provides combinatorial answers to major problems in Lie theoretic representation theory and geometry. Extending such applications to the `singular land’ requires the singular version of the Hecke algebra. Underlying combinatorics of the singular Hecke algebra/category comes from the parabolic double cosets and is the first step in understanding the singular Hecke category. In this talk, I will present a Coxeter theory of the parabolic double cosets developed in a joint work with Ben Elias. In particular, I will explain a generalization of the reduced expressions and describe the braid and non-braid relations.

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