Seminars in Geometric Representation Theory

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Paul Sobaje: Combinatorics for Special Tilting Modules

31st May, 2024

The quest to find a character formula for the simple modules of a reductive algebraic group in positive characteristic took an unexpected turn roughly a decade ago when Williamson found a large number of counterexamples to the Lusztig Conjecture. Since then, the path to the simple characters has gone through the characters of the indecomposable tilting modules, thanks to the work of Riche and Williamson. However, the combinatorics required for determining all tilting characters are quite complicated, and the vast majority of these characters are not necessary to determine the simple characters. This talk is based on our pursuit of a more simplistic model in terms of what we’ve called the 'Steinberg quotient' of special tilting characters.

Pramod Achar: Cleanness in Springer theory

30th May, 2024

In Lusztig's papers from 1985-1986 that invented the theory of character sheaves, he proved (in nearly all cases) a remarkable property of cuspidal perverse Q-sheaves on the nilpotent variety: they are 'clean', meaning that their stalks vanish outside a single orbit. This property is crucial to making character sheaves computable by an algorithm, and it is a precursor of various 'block decompositions' of the derived category studied by various authors (Gunningham, Rider, Russell, and others) later. About 10 years ago, Mautner conjectured that these perverse sheaves remain clean after reduction modulo p (with some exceptions for small p). In this talk, I will discuss the history and context of the cleanness phenomenon, along with recent progress on Mautner’s conjecture.

Charlotte Chan: Generic character sheaves on parahoric subgroups

27th May, 2024

Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the 'generic' part of this desired theory. In the simplest non-trivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne-Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands programme.

Thomas Haines: Pavings of convolution fibres and applications

1st March, 2024

A convolution morphism is the geometric analogue of the convolution of functions in a Hecke algebra. The properties of fibres of convolution morphisms are used in a variety of ways in the geometric Langlands programme and in the study of Schubert varieties. I will explain a very general result about cellular pavings of fibres of convolution morphisms in the setting of partial affine flag varieties, as well as applications related to the very purity and parity vanishing of cohomology of Schubert varieties over finite fields, structure constants for parahoric Hecke algebras, and the (motivic) geometric Satake equivalence.

Vanessa Miemietz: Higher representation theory

12th October, 2023

I will try to motivate the development of a subject called finitary 2-representation theory and explain some techniques and results on the example of Soergel bimodules of finite Coxeter type.

Victor Kac: Unitary representations of minimal W-algebras

1st September, 2023

To each non-zero nilpotent orbit of a simple finite-dimensional Lie superalgebra 𝔤 with a non-degenerate invariant bilinear form one associates a simple vertex algebra, called a quantum affine W-algebra. In the simplest case 𝔤=𝔰𝔩2 one gets the Virasoro vertex algebra.

For the smallest simple Lie superalgebras 𝔤 one gets by this construction all N = 1,2,3,4, and big N = 4 superconformal algebras. I will explain classification of unitary representations of W-algebras, associated to nilpotent orbits of minimal dimension in the even part of 𝔤, which cover all the above examples.

Vera Serganova: Invariant integration on homogeneous superspaces

1st September, 2023

We use Berezin integral in the category of CS-manifolds to construct an invariant integral for the ring of regular functions on a homogeneous affine supervariety G/K. This construction has several applications in representation theory of G. We will explain how it is used in the proof of projectivity detection for support varieties and for description of stable categories for defect 1 supergroups. We also see how this integral can be used to generalize some classical statements from modular representation theory of finite groups to supergroups in characteristic zero.

Peng Shan: Modularity for W-algebras and affine Springer fibres

31st August, 2023

We will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibres. We will also explain how to match the modular group action on the characters with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. This is based on joint work with Dan Xie and Wenbin Yan.

Jethro van Ekeren: Modular tensor categories from exceptional W-algebras

31st August, 2023

I will present results of joint work with T. Arakawa, on representation theory of simple affine W-algebras. For so-called exceptional W-algebras, the category of representations acquires the structure of a modular tensor category, and in this talk I will describe the modular data and fusion rules for some cases. In many cases the modular data matches that of quantum groups at roots of unity, but in other cases, the results are quite mysterious.