Le thème principal de cette conférence sera la Théorie des Représentations (notamment groupes algébriques et groupes finis, groupes de Kac-Moody et algèbres associées), notamment dans ses aspects modulaires et diagrammatiques. Elle sera constituée de 3 mini-cours (4h30 chacun) et d'exposés individuels, et aura lieu dans la station biologique de Besse-et-Saint-Anastaise, le lieu de naissance du groupe Bourbaki.

Modular category O and parity sheaves on flag varieties.

Let G be a semisimple, simply connected group over an algebraically closed field of characteristic p, where p is larger than the Coxeter number for G. Soergel's "modular category O" is a kind of toy model for the representation theory of G. It has only finitely many simple objects, and the combinatorics of the category is controlled by the finite Weyl group, instead of the affine Weyl group. Perhaps its most important feature is that there is a very quick way to get from modular category O to the topology (especially intersection cohomology) of the flag variety. It is in this context that Williamson discovered his famous counterexamples to the Lusztig character formula. A tentative outline for the mini-course is as follows:
Lecture 1: Definition and basic properties of modular category O.
Lecture 2: Brief introduction to parity sheaves on the flag variety.
Lecture 3: Soergel's localization theorem and Williamson's counterexamples.

Representations of finite reductive groups and character sheaves.

These lectures will be focused on the representation theory of reductive groups over finite fields, from a geometric perspective. I will present recent results by Lusztig which allow to consider at the same time:
(a) Representations of the finite reductive group $G(\mathbb{F}_q)$ (of algebraic nature)
(b) Characters sheaves of the reductive group $G$, which are certain $G$-equivariant perverse sheaves on $G$
This will be achieved by constructing (a) and (b) from the same category, the so-called Hecke category of equivariant sheaves on the flag variety, which categorifies the Hecke algebra with equal parameters. As a consequence, we shall see that the irreducible objects in (a) and (b) have the same classification, and that (b) behaves like the limit of (a) at $q=1$.
If time permits, I will explain how one can deduce periodicity properties for the cohomology of Deligne--Lusztig varieties from Lusztig's construction. This is related to how the longest element of the Weyl group acts on the Hecke category.

There are a surprising number of things we still do not know about complex representations of $\mathfrak{sl}_n$. The category is semisimple, and for many people that is the end of the story. However, the category is also monoidal, and the monoidal structure on its morphisms is poorly understood. Understanding this structure is, in turn, crucial for understanding non-semisimple versions of the same category, like modular representations or representations of the quantum group at a root of unity.
The first goal of this lecture series is to explain the diagrammatic algebra of $\mathfrak{sl}_n$-webs, due to Cautis-Kamnitzer-Morrison, which is a monoidal encoding of the morphisms between tensor products of fundamental representations. We will explain some of the key features and uses of this web algebra, such as the double ladders basis, and recursive formulas for clasp idempotents (some of them still conjectural). This is meant to be an understandable survey of the results in arXiv:1510.06840. In fact, the ideas used in this exploration of representations of $\mathfrak{sl}_n$ will generalize to any (nice) generically semisimple monoidal category. The true goal of this lecture series is to explain my philosophy on how to work with such categories. I will discuss how they have the structure of object-adapted cellular categories, with nice cellular bases adapted to the monoidal structure. This gives a somewhat algorithmic way of constructing a presentation and a cellular basis for the category, which we apply to our main example of $\mathfrak{sl}_n$-webs.

A path-theoretic approach to modular simples of symmetric groups

We discuss some new results concerning simple representations of symmetric groups and more general complex reflection groups. We provide new degree-wise upper bounds for all decomposition numbers and provide a homological construction of "completely splittable" simple representations. We generalise these ideas and the familiar notions of generic behaviour and the stability obtained by "tensoring with the determinant" to all cyclotomic Hecke algebras.

On the modular Lusztig--Vogan bijection (joint with P. Achar and S. Riche).

Let $\mathbb{O}$ be a DVR such that its residue field, $\mathbb{F}$, is algebraically closed, let $\mathbb{K}$ be the algebraic closure of its field of fractions, and let $G$ be a (split) simple algebraic group over $\mathbb{O}$. It is known that over both $\mathbb{K}$ and $\mathbb{F}$, there exist bijections between the set of dominant weights, $\mathbf{X}^+$, and the set of irreducible equivariant vector bundles on nilpotent orbits. These bijections are obtained by comparing two t-structures on the bounded derived category of equivariant coherent sheaves on the nilpotent cone. In the $\mathbb{K}$ case, this is known as the "Lusztig--Vogan bijection''. We will refer to the $\mathbb{F}$-version as the "modular Lusztig--Vogan bijection''. In this talk I will explain how these two constructions can be related by a certain base change technique. In general, the reductive quotient of the centralizer of an $\F$-nilpotent element is actually disconnected. However, its module category still has the structure of a highest weight category, which can be naturally induced from the highest weight structure for its connected component. I will also give an overview of this construction, and the role it plays in relating the two bijections.

Self-similarity for positive characteristic Soergel bimodules in affine type.

The diagrammatic category of Soergel bimodules is a linear, additive, monoidal category with deep connections to Kazhdan-Lusztig theory and representation theory. In this talk I will introduce a functor defined over positive characteristic Soergel bimodules for an affine Weyl group which shows that this category exhibits some self-similarity.

Moment graphs, Kazhdan-Lusztig polynomials and combinatorics.

Moment graphs provide a useful incarnation of the Hecke category and allow one to compute Kazhdan-Lusztig polynomials (or their positive characteristic analogue) using only elementary algebra. In the first part of this talk we will review the Braden-MacPherson-Fiebig's theory of sheaves of moment graphs for general Coxeter groups. Then we will focus on the coefficient of q of KL polynomials and on how to interpret it in terms of some simple geometry. In type A this will allow us to give a purely combinatorial formula for the coefficient of q, a result in the direction of the conjectured combinatorial invariance for KL polynomials.

The construction of equivariant vector bundles on projective space.

Motivated by the difficult task of finding low rank indecomposable vector bundles on projective space, we discuss a computational construction strategy for G-equivariant modules over the graded exterior algebra, where G is a finite group. Via an equivariant version of the BGG correspondence, some of these modules can indeed be identified with G-equivariant vector bundles on projective space. For the implementation of our computational construction strategy we use methods of constructive category theory such as a skeletal version of the tensor category of representations of G over a splitting field, and an internalized version of the exterior algebra. These methods are all provided by our computer algebra project CAP (Categories, Algorithms, Programming).

Suppose w is a (reduced) word in an affine Weyl group W. Its shadows with respect to some orientation is the set of all subwords of w that are positive with respect to this orientation. This notion of a shadow generalizes the Bruhat order on W in a natural, geometric way. Similarly one can define shadows of elements in the co-root and weight lattices associated with W.

In this talk I will introduce the concept of shadows and show you a rainbow of applications. We will, for example, learn how to interpret non-emptiness of certain double coset intersections in semisimple algebraic groups using shadows. For example, we will see that one can express various Kostant-type convexity theorems as well as non-emptiness of affine Deligne-Lusztig varieties using shadows. Overall I hope to advertize this geometric approach to the study of double coset intersections.