I will start with some generalities about semi-infinite highest weight categories, following an approach developed in work with Catharina Stroppel. The main example is the category Rep(G) of finite-dimensional representations of a reductive algebraic group. I will also discuss tilting modules and Ringel duality, which relates semi-infinite highest weight categories to semi-infinite lowest weight categories. The latter often have a more combinatorial flavor although they capture all of the same essential structure. This opens the possibility to study the classically important category Rep(G) by non-classical graphical methods which are very different from the traditional algebro-geometric approach.

Another key structural feature of the category Rep(G), and also the subcategory Tilt(G) of tilting modules, is that it is has a tensor product. In order to make sense of this, in the next part of the lecture series, I will introduce some basic notions of monoidal categories, focussing on the classical example of the Schur category which arises naturally from the polynomial part of Rep(GL_n). I will explain how the representation theory of the Schur category can be understood directly by exploiting its natural basis of codeterminants, introduced in a slightly different guise by J. A. Green. I will use this example to motivate the general notion of a triangular basis.

After that we move on to some of the ideas of categorification. The point is to focus not on the category Rep(G) directly, but on certain naturally occuring endofunctors of Rep(G) called translation functors or, more generally, projective functors. This produces new monoidal categories which are themselves incredibly rich. We will focus again on an explicit example: the Heisenberg category which acts naturally on Rep(GL_n) (and also on the closely related categories of representations of symmetric groups). From there, I will magically pass to the Kac-Moody 2-categories of Khovanov, Lauda and Rouquier, following an approach developed in work with Alistair Savage and Ben Webster.

As the name suggests, Kac-Moody 2-categories categorify the usual Kac-Moody algebras, or rather, the corresponding quantum groups. In the last part of the course, I will say more about the representation theory of Kac-Moody 2-categories themselves. Most likely, I will focus instead on a simpler but analogous example arising from the nil-Brauer category introduced in some new work with Weiqiang Wang and Ben Webster. This point of view leads to the idea of a graded triangular basis, which is a surprising new generalization of the triangular bases seen earlier in the course. Finally I hope to explain how the nil-Brauer category is a categorification of the simplest of all \imath-quantum groups, opening up another world of possibilities.

Recommended Reading

Humphreys' "Introduction to Lie algebras and representation theory", Jantzen's "Lectures on quantum groups", Green's "Polynomial representations of GL_n". I will assume good knowledge of the finite-dimensional representation theory of semisimple Lie algebras from Humphreys' book. Some familiarity with the definition of a quantum group from Jantzen's book, and first couple of chapters of Green, especially the definition of the Schur algebra and Schur's product rule, are also recommended but not essential. It may also be worth looking at the first chapter of the book "Monoidal categories and topological field theory" by Turaev and Virelizier, since this gives a gentle introduction to the string calculus which we will often be using to describe monoidal categories diagrammatically. 

The goal of the series will be to explain one of the main problems in the representation of reductive algebraic groups over algebraically closed fields of positive characteristic p, namely the computation of characters of simple modules. This problem has seen a new approach recently, through the computation of characters of a different family of modules, the indecomposable tilting modules, which can be expressed in terms of a new basis of the corresponding affine Hecke algebra, called the p-canonical basis. We will explain the construction of this basis, and some of the main ingredients involved in the approach to this question. 

Recommended Reading

We will assume some familiarity with the basic structure theory of reductive algebraic groups, as presented e.g. in the standard books of Borel, Humphreys and Springer. For a brief introduction to this topic, illustrated by examples, see http://makisumi.com/math/old/reductivegroups.pdf.

The starting point for the course will be the "classical" theory of representations as presented in "Representations of algebraic groups" by Jantzen. (I am planning to recall most of what we will need, but a first exposition to this topic might be useful.)

The course will be based on a book project, which is only partially written at this point, see https://lmbp.uca.fr/~riche/curso-riche-total.pdf. 

Finally, Geordie Williamson has written some very nice surveys on this topic, see https://www.maths.usyd.edu.au/u/geordie/SB1604.pdf and https://arxiv.org/abs/1610.06261.