Date/Time : 24 Jan 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Ziqing Xiang, Southern University of Science and Technology

We introduce a new notion called the quantum wreath product, which produces an algebra $B \wr_Q H(d)$ from a given associative algebra $B$, a positive integer $d$, and a choice $Q = (R, S, \rho, \sigma)$ of parameters. Important examples include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang’s wreath Hecke algebras, Rosso-Savage’s (affine) Frobenius Hecke algebras, Kleshchev-Muth’s affine zigzag algebras, and the Hu algebra that quantizes the wreath product $\Sigma_m \wr \Sigma_2$ between symmetric groups. We will discuss the bases of quantum wreath product algebras, and some of their representations. This is joint work with Daniel Nakano and Chun-Ju Lai.

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Date/Time : 18 Jan 2024 16:30
Venue : S17 #05-12 (SR4)
Speaker : Yongqi Liang, University of Science and Technology of China

Scharaschkin and Skorobogatov conjectured that when smooth projective curves violate the Hasse principle then the violations are explained by the Brauer-Manin obstruction. Colliot-Thelene and Sansuc have similar conjecture for degree 4 del Pezzo surfaces. We use this obstruction to construct explicite algebraic families of degree 4 del Pezzo surfaces and smooth projective curves of genus congruent to 1 modulo 4, such that they violate the Hasse principle. In particular, over every number field,  we find explicite families of elliptic curves whose Tate-Shafarevich groups are non-trivial. This gives an explicite answer to a question of Colliot-Thelene and Poonen asked about 20 years ago, when they proved the existence of such a family of elliptic curves over the field \mathbb{Q} of rational numbers.

This is a joint work with K. Huang. 

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Date/Time : 12 Dec 2023 16:30
Venue : S17 #04-04 (SR3)
Speaker : Lucas Mann, University of Muenster

Based on classical constructions in algebraic topology and algebraic geometry, we introduce abstract 6-functor formalisms, which are a form of “abstract cohomology theories”. It turns out that the abstract theory of 6-functor formalisms is surprisingly rich and has applications in non-geometric fields as well. As an example we provide new tools for studying smooth representations of locally profinite groups — in particular in natural characteristic, where many of the classical representation theoretic methods fail. An application of the theory is a conceptual construction of an anti-involution on the (derived) Iwahori-Hecke algebra, which previously could only be constructed on the associated Ext-algebra using explicit computations.

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Date/Time : 01 Dec 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Chuijia, Technion - Israel Institute of Technology

Kazhdan-Lusztig polynomials play a fundamental role in various Lie theoretical subjects such as Hecke algebras, category O, etc. It is expected that these polynomials only depend on the combinatorial nature of the associated Bruhat intervals. In a recent work of Williamson and his collaborators, they obtain a hypercube decomposition of q derivatives of Kazhdan-Lusztig polynomials via deep geometric machinery, which serves as an important step towards this “combinatorial invariance conjecture”. In this talk, we will try to reinterpret their results from the point of view of parabolic recursions and discuss some applications. This is based on a joint work with Maxim Gurevich.

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Date/Time : 24 Nov 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Fiona Murnaghan, University of Toronto

Let G be a connected reductive group defined over a nonarchimedean local field of odd residue characteristic. Let H be the group of fixed points of an involution of G. A representation of G is said to be H-distinguished if there exists a nonzero H-invariant linear functional on the space of the representation. The H-relatively supercuspidal representations of G are the H-distinguished representations of G whose generalized matrix coefficients are compactly supported modulo HZ, where Z is the centre of G. We will describe results giving a method for constructing H-relatively supercuspidal representations of G.

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Date/Time : 17 Nov 2023 15:00
Venue : S17 #05-11 (SR5)
Speaker : Hiroshi Ishimoto, Osaka Metropolitan University

The original Ibukiyama isomorphism is an isomorphism between vector valued Siegel cusp forms of degree 2 of integral weight and those of half-integral weight. In this talk, I will present a similar isomorphism between vector valued Siegel cusp forms of degree 2 of integral weight and cuspidal automorphic forms on the metaplectic group of degree 2 generating a large discrete series representation at the real place.

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Date/Time : 20 Oct 2023 15:00
Venue : S17 #05-11 (SR5)
Speaker : Apoorva Khare, Indian Institute of Science, Bangalore

We explain recent results involving Schur polynomials – viewed as functions on the positive orthant – which have led to progress along several classical directions of study:

(1) Understanding functions that preserve positive semidefiniteness when applied entrywise to matrices. This journey takes us from work of Schur, Schoenberg, Rudin, and Loewner (1900s) to modern-day results.

(2) The proofs of these recent results lead to characterizing (weak) majorization of real tuples using Schur polynomials; this has since been generalized to all Weyl groups. Majorization inequalities have been studied since Maclaurin and Newton (1700s).

(3) We also explain how all Schur polynomials are lurking within each smooth function, by completing a 1960s determinantal computation of Loewner. This helps extend determinantal identities by Cauchy and Frobenius (1800s) involving one or two geometric series, to all power series, over all unital commutative rings.

(Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.)

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Date/Time : 13 Oct 2023 15:00
Venue : S17 #05-11 (SR5)
Speaker : Yi Shan, Ecole Normake Superieur Paris

Arthur’s conjectures on endoscopic classification enable us to determine whether a representation with a given Arthur parameter occurs in the automorphic spectrum. Chenevier and Renard use Arthur’s results to classify level one automorphic representations for definite orthogonal groups over Q with small ranks and (conjecturally) for anisotropic G_2. In this talk, I will give an overview of their method and then talk about my recent work for the anisotropic semisimple Q-group of type F_4. If time permits, we will discuss the relation of my results with exceptional theta correspondences.

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Date/Time : 13 Oct 2023 16:30
Venue : S17 #05-11 (SR5)
Speaker : Ed Karasiewicz, National University of Singapore

The uniqueness of Whittaker models plays an important role in the representation theory of linear reductive groups due to its relation to L-functions. However such uniqueness fails in general for nonlinear covering groups. We investigate this failure of uniqueness through the Gelfand-Graev representation, which is the dual of the Whittaker space.

Using the pro-p Iwahori-Hecke algebra, we describe the Iwahori-fixed vectors in the Gelfand-Graev representation of covering groups as a module over the Iwahori-Hecke algebra, generalizing work of Barbasch-Moy and Chan-Savin for linear groups. As an application we 1) compute the dimension of the space of Whittaker models for constituents of unramified principal series for regular and unitary characters; 2) describe a connection to quantum affine Schur Weyl duality in the case of covers of GL(r).

This is joint work with Fan Gao and Nadya Gurevich.

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