Geometric Representation Theory
Hessenberg varieties are a distinguished family of projective varieties associated to a semisimple complex algebraic group. We use the formalism of perverse sheaves to study their cohomology rings. We give a partial characterization, in terms of the Springer correspondence, of the irreducible representations which appear in the action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety.
Categorical representation theory is filled with graded additive categories (defined by generators and relations) whose Grothendieck groups are algebras over \mathbb{Z}[q,q^{-1}]. For example, Khovanov-Lauda-Rouquier (KLR) algebras categorify the quantum group, and the diagrammatic Hecke categories categorify Hecke algebras. Khovanov introduced Hopfological algebra in 2006 as a method to potentially categorify the specialization of these \mathbb{Z}[q,q^{-1}]-algebras at q = \zeta_n a root of unity. The schtick is this: one equips the category (e.g.
For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan–Lusztig cells in the affine Weyl group. In this talk, I will review these results, and I will explain a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys. This is joint work with W. Hardesty and S. Riche.
Koszul duality, as conceived by Beilinson-Ginzburg-Soergel, describes a remarkable symmetry in the representation theory of Langlands dual reductive groups. Geometrically, Koszul duality can be stated as an equivalence of categories of mixed (motivic) sheaves on flag varieties. In this talk, I will argue that there should be an an 'ungraded' version of Koszul duality between monodromic constructible sheaves and equivariant K-motives on flag varieties. For this, I will explain what K-motives are and present preliminary results.
Admissible representations of real reductive Lie groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lustig—Vogan in the 80’s in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I’ll describe a categorification of this module using Soergel bimodules, with a focus on examples. This is work in progress.
Suzuki's functor relates the representation theory of the affine Lie algebra to the representation theory of the rational Cherednik algebra in type A. In this talk, we discuss an extension of this functor to the critical level, t=0 case. This case is special because the respective categories of representations have large centres. Our main result describes the relationship between these centres, and provides a partial geometric interpretation in terms of Calogero-Moser spaces and opers.
In this note we give an alternative presentation of the rational
Cherednik algebra H_c corresponding to the permutation representation of
S_n. As an application, we give an explicit combinatorial basis for all
standard and simple modules if the denominator of c is at least n, and
describe the action of H_c in this basis. We also give a basis for the
irreducible quotient of the polynomial representation and compare it to
the basis of fixed points in the homology of the parabolic Hilbert
The category of Harish-Chandra bimodules is ubiquitous in representation theory. In this talk I will explain their relationship to the theory of dynamical R-matrices (going back to the works of Donin and Mudrov) and quantum moment maps. I will also relate the monoidal properties of the parabolic restriction functor for Harish-Chandra bimodules to the so-called standard dynamical R-matrix. This is a report on work in progress, joint with Artem Kalmykov.
The preprojective algebra of a quiver naturally appears when computing
the cotangent to the quiver moduli, via the moment map. When considering
the derived setting, it is replaced by its differential graded (dg)
variant, introduced by Ginzburg. This construction can be generalized
using potentials, so that one retrieves critical loci when considering
moduli of perfect modules.
Our idea is to consider some relative, or constrained critical loci,
deformations of the above, and study Calabi--Yau structures on the
I will explain a generalized Albanese property for smooth curves, which implies Deligne's geometric class field theory with arbitrary ramification. The proof essentially reduces to some well-known Cartier duality statements. This is joint work with Andreas Hayash.