# Summer School on Geometric Representation Theory

9 July—13 July 2018, Institute of Science and Technology Austria

# Titles and abstracts

## Mini-courses

### Iain Gordon — Symplectic Reflection Algebras: the KZ functor and Quiver Varieties

I will discuss the construction of Symplectic Reflection Algebras generally, the rational Cherednik algebras and the geometric and algebraic construction of the KZ functor through a relationship to quantization of certain quiver varieties. More generally, if I have time, I will also discuss relations between these and resolution of symplectic singularities, where quiver varieties are now local models.

### Andrei Negut — Moduli spaces of sheaves: geometry and representation theory

The purpose of this mini-course is to study moduli spaces of sheaves on a surface, as well as Hecke correspondences between them. Hecke correspondences are varieties parametrizing several sheaves on the surface which only differ at points, and they lead to interesting operators on the cohomology, Chow groups and K-theory of the moduli spaces of sheaves. Applications of this picture range from mathematical physics to classical problems in the algebraic geometry of hyperkähler manifolds.
Lecture notes are available here (pdf).

### Nicholas Proudfoot — Category $\mathcal{O}$, symplectic duality, and the Hikita conjecture

The category $\mathcal{O}$ for a semisimple Lie algebra can be interpreted geometrically in terms of the Springer resolution, and then generalized to other symplectic resolutions. These resolutions tend to come in dual pairs whose associated categories are related by Koszul duality. After introducing these concepts, I will discuss various versions of the Hikita conjecture, which relates the cohomology of one resolution to the coordinate ring of its dual.
Lecture notes and exercises are available on Nick’s webpage.

### Catharina Stroppel — Springer fibers: basic properties and applications to categorification

I will explain a few basic properties of (finite) Springer fibers and indicate how they show up in categorifications. The focus will be on explicit calculations, (partly outside type $A$) on the one hand and some general philosophy behind the related categorifications on the other hand.

## Research talks

### Tina Kanstrup — Link invariants and matrix factorizations.

Khovanov and Rozansky defined a link invariant called triply graded homology. It is conjectured by Gorsky, Negut and Rasmussen that this invariant can be expressed geometrically by a functor from complexed of Soergel bimodules to the derived category of coherent sheaves on the (dg) flag Hilbert scheme followed by taking global sections. A functor with similar properties has been constructed by Oblomkov and Rozansky using matrix factorizations and it is believed that this functor solves the conjecture. The aim of this joint work in progress with Roman Bezrukavnikov is to relate the two constructions using previous work of Arkhipov and Kanstrup.

### Ivan Mirković — Loop Grassmannians of quivers

This is a joint work with Yaping Yang and Gufang Zhao. An earlier construction generalizes the construction of loop Grassmannians by replacing the data of a reductive group by a based quadratic form. The talk will recast this construction in terms of quivers and use this to quantize the construction.

### Neil Saunders — Representations and Combinatorics of Weyl Groups from Springer Theory

Let $G$ be a connected, reductive algebraic group defined over $\mathbb{C}$. The celebrated Springer Correspondence gives a bijection between theirreducible representations of the Weyl group $W$ of $G$ and certain pairs comprising a $G$-orbit on nilpotent cone of the Lie algebra of $G$ and an irreducible local system attached to that $G$-orbit. These irreducible representations can be concretely realised as a W-action on the top degree homology of the so-called Springer fibres. These Springer fibres are geometrical very rich and provide interesting Weyl group combinatorics: for instance, the irreducible components of these Springer fibres form a basis for the corresponding irreducible representation of $W$. In this talk, I’ll give a general survey of the Springer Correspondence and discuss recent joint work on Kato’s Exotic Springer correspondence in Type C.

### Alexander Shapiro — On modular functor and representation theory.

A quantized higher Teichmüller theory assigns an algebra and its Hilbert space representation to each choice of a surface and a Lie group $G$. The modular functor conjecture, as formulated by Fock and Goncharov, predicts how this assignment behaves under cutting and gluing of surfaces. I will outline a proof of this conjecture for $SL_n$ and discuss how different statements on positive representations of quantum groups arise as particular cases of the conjecture. This is a joint work with Gus Schrader.

### Tom Sutherland — Theta functions on moduli spaces of local systems

We consider the so-called theta basis of the ring of regular functions on moduli spaces of $SL(2,\mathbb{C})$-local systems on Riemann surfaces with (possibly irregular) singularities, whose structure coeﬃcients come from the tropical geometry of the moduli space which here are given by the representation theory of an associated quiver. We will focus on the lowest dimensional examples of such moduli spaces for which three elements of the theta basis which compute the trace of the monodromy of the local systems around particular loops on the Riemann surface embed the moduli space as an affine cubic surface in $\mathbb{C}^3$. In these cases the coefficients of the deﬁning cubic equation arise from the representation theory of a ﬁnite, affine or elliptic Dynkin quiver.