TCC seminar series: Representation Theory and Lie Theory
Date & time: The course will start on Monday 25th of April 2022, 10:00-12:00. It will run weekly via Teams for 8 weeks.
Monday 2nd of May is a Bank Holiday, so the session will be on Wednesday 4th, 10:00-12:00.
The course is designed for grad students at participating universities (Bath, Bristol, Imperial, Oxford, Warwick).
Prerequisites: I have tried to keep prerequisites to a minimum, so the course should be accessible for PhD students at any stage of study. A solid grounding in algebra is required. Experience in algebraic geometry, representation theory and category theory would be helpful.
Course notes: Download here.
Summary: The following is an ambitious overview of what I plan to discuss within the 16 hours of lectures. We will definitely not have time to cover all of this material but, we'll start at the beginning and see how far we get.
Representation theory of Lie algebras lies at the very core of pure mathematics, having fundamental connections to geometry, mathematical physics, but the foundational methods came from ring theory and abstract algebra. The course will begin by explaining the fundamental problems and methods in representation theory of Lie algebras and enveloping algebras, focusing on irreducible representations and their primitive ideals. We highlight the structures which are common to all Lie algebras and highlight the differences and similarities between the ordinary and modular setting.
Some of the most important examples of topological groups in representation theory, both for the sake of applications and for their guiding influence within the field, are algebraic groups. Each algebraic group gives rise to a Lie algebra and the most interesting examples of algebraic groups are the reductive groups and semisimple groups, which are very closely related. The most remarkable result here, due to Chevalley and Demazure, is that semisimple and reductive groups are classified by combinatorial data independent of the characteristic of the field. Thus they give rise to a family of Lie algebras over algebraically closed fields of every characteristic. The second part of this course will sketch this classification process in broad strokes, and then examine the canonical example of a reductive group in detail: the general linear group.
In the final part of the course we will survey the representation theory of Lie algebras of reductive groups in both zero and positive characteristics. When g = Lie(G) for a reductive group over a field of characteristic zero the classical approach to constructing finite dimensional representations is highest weight theory, which leads naturally to Bernstein-Gelfand-Gelfand (BGG) category O, which contains many interesting infinite dimensional modules, related by beautiful combinatorics. From here we can construct all primitive ideals (Duflo's theorem) and we will attach a certain invariant to each such ideal, known as the associated variety. By a theorem of Jospeh it is the closure of a nilpotent orbit.
Over fields of characteristic p > 0 (with some small restrictions on p) the representation theory of Lie algebras of reductive groups can also be constructed using an analogue of highest weight theory, but there are significant differences to the complex case, due to the presence of the p-centre, a large central subalgebra of the enveloping algebra. Nonetheless, the process leads to a total classification of all irreducible modules for the general linear Lie algebra, and a partial classification in other types.
The state-of-the-art of the field (possibly a subject of a future course):
The problem of understanding composition multiplicities in category O was one of the longest standing open problems in representation theory, and led to the formulation of the Kazhdan-Lusztig conjectures. These were eventually proven by Beilinson-Bernstein and Brylinski-Kashiwara (independently) using the localisation theorem: the category of g-modules is equivalent to the category of quasicoherent D-modules over the flag variety. This landmark theorem was the birth of geometric representation theory, and we could try to sketch some of the main ideas in the proof.
Over fields of positive characteristics, there is a localisation theorem due to Bezrukavnikov-Mirkovic-Rumynin, which relates the derived categories of g-modules to the derived category of D-modules, both with a fixed generalised central character. This leads to a relationship between g-modules to the derived category of coherent sheaves on the Springer fibre which, in principle, allows us to count the number of simple modules. One day I hope to explain these facts in more detail.
A modern approach to the representation theory of these Lie algebras (in any characteristic) involves the use of a certain quantum Hamiltonian reduction, known as the finite W-algebra. We attach a W-algebra to each nilpotent orbit in the underlying Lie algebra, and they lead us to a new family of g-modules, known as Whittaker modules. Over fields of positive characteristic Whittker modules include all simple modules, whilst over \C they form a proper subcategory, quite different to BGG O. Finite W-algebras first arose in modular representation theory as Premet refined his proof of the second Kac-Weisfeiler conjecture, and now they play a key role in the classification of primitive ideals in enveloping algebras of complex semisimple Lie algebras, efficiently organising primitive ideals in terms of their associated varieties.
PhD project:
I have PhD several projects in mind, candidates should write to me to discuss lt803@bath.ac.uk.
Ellis Caird (2021-2022) - Classification of complex semisimple Lie algebras
James Martin (2023-2024) - Representation theory of the symmetric group (thesis).
Rober Leonard (2023-2024) - Representation theory of complex semisimple Lie algebras