Department of Mathematical Sciences > Research > Conferences > Kazdan-Lusztig Theory

## Masterclass on

# Kazdan-Lusztig theory: Background, recent developments and open problems

## Sept 4+11+13, 2013

**Content:** 3 Lectures by Eric Marberg (Stanford) on: Kazdan-Lusztig theory. Background, recent developments, and open problems.

**Abstract**: This will be a series of three lectures focused on the Iwahori-Hecke

algebra of a Coxeter system, and in particular its distinguished

Kazhdan-Lusztig basis. The talks will aim to be accessible to graduate

students with some background in group representations.

The first lecture, after giving some background on Coxeter groups, will

review the classical proofs of the existence of the Iwahori-Hecke algebra

of any Coxeter system, and of the Kazhdan-Lusztig basis of this algebra.

We will discuss the sense in which the Kazhdan-Lusztig basis provides an

example of a "canonical basis" and review the major conjectures related to

the KL basis.

The second lecture will provide a basic introduction to the category of

Soergel bimodules. After defining this category and exploring some

examples, we will discuss how Soergel bimodules serve as a

categorification the KL basis of the Iwahori-Hecke algebra. We will then

give a very brief survey of some recent work of Elias and Williamson using

Soergel bimodules to prove positivity properties of the Kazhdan-Lusztig

basis.

The final lecture will discuss some sources of open problems related to

Kazhdan-Lusztig theory, where variations of the new methods described in

the second lecture might find application. Some possible topics, time

permitting, include: still open positivity conjectures related to the KL

basis; the combinatorial invariance conjecture; analogues of "KL bases"

for modules of the Hecke algebra and their positivity properties;

Stembridge's finiteness results for admissible W-graphs; Lusztig's

heuristic definition of the "unipotent characters" of finite Coxeter

systems.

**Notes**: