Titres et résumés

Peter Fiebig: Arithmetic challenges on Bruhat graphs

We discuss various instances where certain rational algorithms on Bruhat graphs produce a set of exceptional prime numbers. These sets are very poorly understood and are very closely linked to the set of characteristics for which Lusztig's modular character conjecture fails.

Michael Finkelberg: Double affine Grassmannian

This is a joint project with A.Braverman. We develop an affine analogue (for an affine Kac-Moody group) of the geometric Satake isomorphism. At the present moment, there are more conjectures and open questions than theorems.

Joel Kamnitzer: Yangians and quantizations of slices in the affine Grassmannian

We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. I will also discuss recent results concerning the representation theory of these quotients.

Paul Levy: Singularities in nilpotent cones of exceptional Lie algebras

It is well known that the set of nilpotent orbits for the action of a simple complex algebraic group on its Lie algebra is finite, and that each nilpotent orbit equals the smooth locus of its own closure. Given a pair (O, O') of nilpotent orbits such that O' is contained in the closure of O (called a degeneration), one obtains an associated singularity by considering the (very) local geometry of the closure of O near a point of O'.

In the classical cases, these singularities were studied by Kraft and Procesi in a series of papers in the late 1970s and early 1980s. Their work establishes various equivalences between the singularities associated to degenerations in classical Lie algebras. Subsequently Kraft studied the nilpotent cone of a Lie algebra of type G2. However, until recently there has been little work on the singularities associated to degenerations in the other exceptional types. In this talk I will outline the results of some recent research on types E and F. This is joint work with Baohua Fu, Daniel Juteau and Eric Sommers.

Carl Mautner: Modular perverse sheaves on nilpotent cones

This talk will address some structural results about the category of modular perverse sheaves on nilpotent cones. In particular, I will describe the existence of a natural autoequivalence on the equivariant derived category and will explain how it is analogous to an algebraic equivalence, namely Ringel duality for Schur algebras. This is joint work with Pramod Achar.

Olaf Schnürer: Matrix factorizations, semi-orthogonal decompositions and motivic measures

I will report on joint work with Valery Lunts. We establish semi-orthogonal decompositions for matrix factorizations on blowing-ups and use them for constructing motivic measures with values in Grothendieck rings of saturated dg categories.

Toshiaki Shoji: Character sheaves on exotic symmetric spaces

It is well known that there exists a natural bijection between the set of nilpotent orbits of GL_n and the set of irreducible representations of the symmetric group S_n through the intersection cohomology on the closure of nilpotent orbits (Springer correspondence). Also it is known that Kostka polynomials are interpreted geometrically in terms of such intersection cohomologies.

Let V be a 2n-dimensional vector space, and put G = GL(V), K = Sp(V). We consider the variety X = G/K \times V, with the diagonal action of K. Since its "unipotent part" X_0 is isomorphic to the exotic nilpotent cone introduced by Kato, we call X the exotic symmetric space. In this series of talks, we discuss about the theory of character sheaves on X. Based on it, we show that the analogy of the Springer correspondence holds between the set of K-orbits in X_0 and the set of irreducible representations of the Weyl group of type C_n, and show that the intersection cohomology of those K-orbits gives a geometric interpretation of some (generalized) Kostka polynomials.

Our construction of character sheaves on X is based on the explicit data. There exists, however, a more conceptual definition of character sheaves on X based on the idea of Ginzburg in the case of symmetric spaces. We show that those two definitions actually coincide, which implies the classification of Ginzburg type character sheaves.

Geordie Williamson: The Hodge theory of Soergel bimodules

Soergel bimodules provide the most explicit incarnation of the Hecke category. I will start by explaining how Soergel bimodules arise as the equivariant (hyper)cohomology of semi-simple complexes on the flag variety. This will serve two purposes: it will (hopefully) justify the subject matter at a conference on perverse sheaves; and it make the later appearance of certain rather deep Hodge theoretic properties less surprising. I will then outline a beautiful package of ideas due to de Cataldo and Migliorini, which gives an inductive road map for establishing these Hodge theoretic properties, along with the decomposition theorem. Finally, I will explain joint work with Ben Elias where we establish these properties algebraically. This gives a well-behaved "Hecke category" for any Coxeter system. If time permits I will discuss work in progress on the "non semi-small case".