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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Date/Time : 06 Oct 2021 15:00
Venue : via Zoom
Speaker : Jan Frahm, Aarhus University

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https://nus-sg.zoom.us/j/81603760900?pwd=d0lGRktVY0hJSTM5ODBCWmZmTVRrUT09

Meeting ID: 816 0376 0900
Passcode: 883638

Abstract:  ranching problems ask for the behaviour of the restriction of an irreducible representation of a group $G$ to a subgroup $H$. In the context of smooth representations of real reductive groups, this typically leads to the study of multiplicities with which an irreducible representation of $H$ occurs as a quotient of an irreducible representation of $G$. Here, both quantitative results such as multiplicity-one theorems and qualitative results such as the Gan-Gross-Prasad conjectures are of interest. In the context of unitary representations of real reductive groups, one can go a step further and explicitly decompose in irreducible representation of $G$ into a direct integral of irreducible representations of $H$. I will explain how branching laws for unitary representations are related to those in the smooth category, and how one can use an analytic continuation procedure along a principal series parameter to obtain explicit branching laws from certain Plancherel formulas for homogeneous spaces.

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Date/Time : 29 Sep 2021 14:00
Venue : via Zoom
Speaker : Shunsuke Yamana, Osaka City University

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https://nus-sg.zoom.us/j/86263979122?pwd=WXNuYUdnNXdhQkNOODJJNlB3MnRhQT09

Meeting ID: 862 6397 9122
Passcode: 630599

Abstract:  p-adic L-functions involve modified p-factors which measure the discrepancy between the p-adic and complex L-values in the interpolation formula. It is a puzzling fact that this factor can vanish at the central point. Then the p-adic L-function trivially vanish at the point, and such a zero is called an exceptional zero. The p-adic L-function of an elliptic curve has an exceptional zero if and only if it has split multiplicative reduction at p, and the precise relation between derivative of the p-adic L-function and the algebraic part of the complex L-value was conjectured by Mazur-Tate-Teitelbaum and proved by Greenberg-Stevens. 　 There have been many attempts to extend this result of Greenberg-Stevens to more general automorphic forms.

In this talk I will consider the exceptional zeros of the cyclotomic twisted triple product p-adic L-function associated to elliptic curves over rationals and a real quadratic field, and prove an identity between derivatives of the p-adic L-function and complex L-values. I will also consider exceptional zeros of a certain p-adic L-function of degree 6 associated with two rational elliptic curves.

This is a joint work with Ming-Lun Hsieh.

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Date/Time : 22 Sep 2021 14:00
Venue : via Zoom
Speaker : Hideya Watanabe, Osaka City University

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https://nus-sg.zoom.us/j/82636421526?pwd=WUhpZzJPMW1mWXJtRkhtZHRFeVBOUT09

Meeting ID: 826 3642 1526
Passcode: 189069

Abstract: In the representation theory of a quantum group $U$ of finite
type, a based module means a $U$-module with a canonical and crystal
basis. One of the most important properties of canonical bases is
cellularity. In this talk, I generalize the notion of based modules to the
representation theory of the i-quantum group of type AI, and explain that
the associated canonical bases are cellular. Taking the crystal limit of
the canonical bases, a new combinatorial model for irreducible modules
over the orthogonal group is obtained.

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