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
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