# Workshop on Representation Theory and Geometry

### The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras

Gurbir Dhillon

Let g be a semisimple Lie algebra. The affine W-algebra associated to g is a topological algebra which quantizes the algebraic loop space of the Kostant slice. It is constructed as a quantum Hamiltonian (alias quantum Drinfeld--Sokolov) reduction of...

### Flag manifolds over semifields II

The study of totally positive matrices, i.e., matrices with positive minors, dates back to 1930s. The theory was generalised by Lusztig to arbitrary split reductive groups using canonical bases, and has significant impacts on the theory of cluster...

### Towards derived Satake equivalence for symmetric varieties

In an ongoing project of D. Ben-Zvi, Y. Sakellaridis and A. Venkatesh, the authors propose a conjectural generalization of the derived Satake equivalence for complex reductive groups to spherical varieties. I will describe a program aimed at...

### Double covers of tori and the local Langlands correspondence

Given a maximal torus T of a connected reductive group G over a local field F, there does not exist a canonical embedding of the L-group of T into the L-group of G. Generalizing work of Adams and Vogan in the case F=R, we will construct a natural...

### Lax 2-tensor constructions

In joint work with Andy Manion, we required a generalization of the tensor product of 2-representations in order to reconstruct the 2-dimensional part of Lipshitz-Oszvath-Szabo’s bordered Heegaard-Floer theory. I will discuss this and possible...

### Representations of p-adic groups

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the local Langlands program is...

### Stokes phenomena, Poisson-Lie groups and quantum groups

Let G be a complex reductive group, G? its dual Poisson-Lie group, and g the Lie algebra of G. G-valued Stokes phenomena were exploited by P. Boalch to linearise the Poisson structure on G?. I will explain how Ug-valued Stokes phenomena can be used...

### A Hecke action on the principal block of a semisimple algebraic group

I will explain the construction of an action of the Hecke category on the principal block of representations of a connected reductive algebraic group over an algebraically closed field of positive characteristic, obtained in joint work with Roman...

### Rationality properties of complex characters of finite groups

What can one say about the fields of values of irreducible complex characters ?? of a given finite group GG? In particular when GG is a finite (quasi)simple group? What about the McKay situation'', i.e. when the degree of
?? is coprime to a fixed...

### New isolated symplectic singularities with trivial fundamental group

In 2000, Arnaud Beauville introduced the notion of symplectic singularities and raised the question of classifying isolated symplectic singularities with trivial local fundamental group: the latter condition is meant to avoid the numerous quotient...