Representations of groups of Lie type

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.

Jon Carlson: Endotrivial modules for finite groups of Lie type

27th May, 2024

Suppose that G is a finite group and that k is a field of characteristic p > 0. A kG-module M is an endotrivial module if Hom_k(M,M) ≅ M^∗ ⊗ M ≅ k ⊕ (proj). The endotrivial modules form the Picard group of self-equivalences of the stable category and have been classified for many families of groups. In this lecture I will describe some progress in the classification of endotrivial kG-modules in the case that G is a group of Lie type. We concentrate on the torsion subgroup of the group endotrivial modules, as the torsion free part was determined in earlier work. The torsion part consists mainly of modules whose restrictions to the Sylow subgroup of G are stably trivial. In most cases such modules have dimension 1, but the exceptions are notable.

Jessica Fintzen: An introduction to representations of p-adic groups

25th March, 2024

An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool not just within representation theory, but also for the construction of an explicit and a categorical local Langlands correspondence, and has applications to the study of automorphic forms, for example. In my talk I will introduce p-adic groups and explain that the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will then provide an overview of what we know about the structure of these Bernstein blocks. In particular, I will sketch how to use a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

Naihuan Jing: Characters of GL_n(q) and vertex operators

3rd March, 2024

Irreducible characters of the finite group GL_n(q) were determined by Green in a remarkable paper that has influenced representation theory greatly. In this talk, I will discuss a vertex algebraic approach to construct and compute all complex irreducible characters of GL_n(q). Green's theory is recovered and enhanced under the realization of the Grothendieck ring of representations R(G)=⨁_n≥0R(GL_n(q)) as two isomorphic Fock spaces. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. This offers a simplification to identify the Fock space R(G) as the Hall algebra of symmetric functions. We will also discuss how to compute the characters in general.

Emily Norton: Decomposition numbers for unipotent blocks with small 𝔰𝔩₂-weight in finite classical groups

24th October, 2023

There are many familiar module categories admitting a categorical action of a Lie algebra. The combinatorial shadow of such an action often yields answers to module-theoretic questions, for instance via crystals. In proving a conjecture of Gerber, Hiss, and Jacon, it was shown by Dudas, Varagnolo, and Vasserot that the category of unipotent representations of a finite classical group has such a categorical action. In this talk I will explain how we can use the categorical action to deduce closed formulas for certain families of decomposition numbers of these groups.

Pavel Turek: On stable modular plethysms of the natural module of SL₂(𝔽_p) in characteristic p

24th January, 2023

To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field k of non-zero characteristic. However, by describing the representation ring of kSL₂(𝔽_p) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of kSL₂(𝔽_p) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural kSL₂(𝔽_p)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are 'close' to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.

Geordie Williamson: Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics

27th June, 2022

This will be a course on the representation theory of algebraic groups, and relations to the representation theory of symmetric groups. Reductive algebraic over finite fields and their algebraic closures are fascinating objects: one the one hand they look like Lie groups, but on the other hand they look like finite groups. Thus they mix two very different areas of mathematics. I will outline some of the basic theory, and then move on to questions of current interest.

Ting Xue's lectures in the previous week will provide essential background. I will aim to point out connections to the modular representation theory of finite groups. Although not essential, some background in algebraic geometry (e.g. the first three chapters of Hartshorne's Algebraic Geometry) will help with understanding latter parts of the course.

Inna Entova-Aizenbud: Representation stability for GL_n(𝔽_q)

10th May, 2022

I will present some results from a work in progress joint with Thorsten Heidersdorf on the Deligne categories for the family of groups GL_n(𝔽_q), for non-negative integers n. The Deligne categories interpolate the tensor categories of complex representations of GL_n(𝔽_q), and have been previously constructed by F. Knop and E. Meir (for certain values of n). I will describe some properties of these categories as well as their relation to the category of algebraic representations of the infinite group GL_∞(𝔽_q).

Michael Larsen: Character estimates for classical finite simple groups

27th May, 2021

This is intended to complement the recent talk of Pham Huu Tiep in this seminar but will not assume familiarity with that talk. The estimates in the title are upper bounds of the form |χ(g)|≤χ(1)α, where χ is irreducible and α depends on the size of the centralizer of g. I will briefly discuss geometric applications of such bounds, explain how probability theory can be used to reduce to the case of elements g of small centralizer, discuss the level theory of characters, and conclude with the reduction to the case of characters χ of large degree. For such pairs (g,χ), exponential character bounds are trivial.

Tag - Representations of groups of Lie type