Workshop on Lie Groups, Lie Algebras and their Representations

Abstract. Perverse sheaves on the affine Grassmannian of a reductive group $G$ encode a great deal of representation-theoretic information. In characteristic 0, these sheaves have been studied in depth since the 1990s. In this talk, I will discuss recent advances in the positive-characteristic case, including the proof of the Mirkovic-Vilonen conjecture and the relationship with the Springer resolution of the Langlands dual group. This is joint work with L. Rider. I will also explain the connection to closely related independent work of Mautner-Riche.

Abstract. We calculate the result of the Drinfeld-Laumon construction applied to the irreducible direct summands in the Springer-Laumon sheaf for the trivial local system on a curve. Motivation comes from Bezrukavnikov's conjectures in representation theory of the rational Cherednik algebra in characteristic p and from geometric Langlands duality for the trivial local system.

Abstract. For algebraic groups $G$ of "exponential type" over an algebraically closed field of positive characteristic, we have introduced a "support variety" for each rational $G$-module. Our definition involves 1-parameter subgroups of $G$ and relates to constructions of Suslin-Friedlander-Bendel for Frobenius kernels of $G$. We have also introduced a filtration on rational $G$-modules for such $G$ adapted to these support varieties. We present work in progress to give a cohomological interpetation of these support varieties for $G$ unipotent which conveys some understanding of our general constructions.

Abstract. Many important algebraic objects such as quantum groups, Hecke algebras, and Heisenberg algebras admit diagrammatically defined categorifications. These categorifications are certain monoidal categories defined by planar diagrams modulo local relations. In this talk we will describe two natural choices of `decategorification' map - the 'Grothendieck group' and the 'trace' - each of which transform categories back into algebras. We will show that these two constructions are related by an algebraic analog of the Chern character map. In particular, when these graphical categories are represented in a geometric context (like quiver varieties, flag varieties, or Hilbert schemes), the decategorification procedures and their relationship exactly correspond to the relationship between cohomology and K-theory expressed by the (geometric) Chern character map. Hence, 'Grothendieck group' and 'trace' provide completely algebraic/diagrammatic way to see the relationship between pairs such as quantum groups/current algebras, Hecke algebras/smash product algebras, Heisenberg algebras/$W$-algebras, and conjecturally Loop algebras/Yangians.

Abstract. Nakajima quiver varieties are moduli spaces of certain representations of quivers. They play an important role in Algebraic Geometry, Mathematical Physics and Geometric Representation theory. Their quantizations are noncommutative associative algebras with interesting and rich representation theory conjecturally related to deep geometric properties of the underlying varieties. I will explain some reasons to be interested in that representation theory and also some results in the important special case of quantized Gieseker moduli spaces based on http://arxiv.org/abs/1405.4998. All necessary information about quiver varieties and their quantizations will be introduced during the talk.

Abstract. Motivated by questions in representation theory, Carlson, Friedlander and the speaker instigated the study of projective varieties of abelian $p$-nilpotent subalgebras of a fixed dimension $r$ for a $p$-Lie algebra $\mathfrak g$. These varieties are close relatives of the much studied class of varieties of $r$-tuples of commuting $p$-nilpotent matrices which remain highly mysterious when $r>2$. In this talk, I will present some of the representation-theoretic motivation behind the study of these varieties and describe their geometry in a very special case when it is well understood: namely, when $r$ is the maximal dimension of an abelian $p$-nilpotent subalgebra of $\mathfrak g$ where $\mathfrak g$ is a Lie algebra of a reductive algebraic group. This is joint work with J. Stark.

Abstract. Starting with the work of Khovanov, there has been a lot of interest in recent years in constructing strong categorifications of the Heisenberg algebra and various analogues. Using diagrammatic techniques, we will present a general framework of strong categorification that depends on a given (graded) Frobenius algebra and discuss how to recover previous results of Cautis-Licata and Hill-Sussan for particular choices of the Frobenius algebra. This is work in progress, joint with Alistair Savage.

Abstract. I will describe a new way to study the structure of certain affine Deligne-Lusztig varieties. In particular, I will explain how the Langlands dual group appears in the parametrization of irreducible components of these varieties.