Week 1:
Joel Kamnitzer (University of Toronto)
Introduction to geometric representation theory
As an introduction to geometric representation theory, I will present
three different constructions of the irreducible representations
of GL_n. All three constructions involve the geometry of the flag
variety and related varieties. We will begin with the most ``classical''
construction -- the Borel-Weil theorem, which constructs representations
as sections of line bundles on the flag variety. Next, we will consider
Ginzburg's construction of representations using the homology of
partial Springer fibres. Finally, we will look at the geometric
Satake correspondence, where we realize representations using the
intersection homology of orbit closures in the affine Grassmannian.
All our constructions generalize to all complex reductive groups,
but we will focus on GL_n in order to simplify the notation and
to make the geometry more transparent.
It would be helpful if the participants know the representation
theory of GL_n (classification of representations by highest weight)
and some of the geometry of the flag variety, but these topics will
be reviewed during the course.
Seaok-Jin Kang (Seoul National University)
Introduction to quantum groups and crystals
In this course, we will give an introduction to the theory of quantum
groups and crystal bases. We will start with the basic theory of
Kac-Moody algebras and their quantum deformation. We then move on
to the crystal basis theory including tensor product rule, abstract
crystals and global bases. As an illustration, the crystal bases
for quantum general linear algebras will be realized as the crystals
consisting of semistandard Young tableaux. Our main objective is
to study the perfect crystals for quantum affine algebras. Using
the fundamental crystal isomorphism theorem, we will give realizations
of irreducible highest weight crystals in terms of Kyoto paths.
Finally, we will discuss combinatorics of Young walls. The Young
walls consist of colored blocks with various shapes and can be regarded
as generalizations of Young diagrams. The irreducible highest weight
crystals for classical quantum affine algebras will be realized
as the affine crystals consisting of reduced Young walls.
Prerequisites: I will define and review Kac-Moody algebras and their
representation theory. I will assume that students are familiar
with the materials in the 1st year graduate algebra course, e.g.
rings, modules, tensor product, but no more than that.
Erhard Neher (University of Ottawa)
Affine, toroidal and extended affine Lie algebras
The course will give an introduction to generalizations of affine
Lie algebras. The focus will be on extended affine Lie algebras,
whose structure theory will be presented: root systems, construction
of extended affine Lie algebras in terms of generalized loop algebras,
the so-called Lie tori. To illustrate the general theory we will
present many examples, like toroidal algebras and extended affine
Lie algebras associated to quantum tori and multiloop algebras.
Prerequisites: Basic theory of simple finite-dimensional Lie algebras.
Some understanding of affine Lie algebras will be helpful, but not
required.
WEEK 2:
Weiqiang Wang (University of Virginia)
Nilpotent orbits and W algebras
In recent years, the (finite) W algebras associated to a simple
Lie algebra $g$ and a nilpotent element of $g$ (which can be traced
back to Kostant and Lynch) have been studied intensively by many
people from different viewpoints including Premet, Skryabin, Brundan,
Kleshchev, Gan, Ginzburg, an others. This mini-course is aimed at
non-experts with a good background on basic Lie theory. We shall
present some basic constructions, connections, and applications
of finite W algebras over the complex field, with particular focus
on type A, such
as:
-- various equivalent definitions of W algebras.
-- independence of W algebras on good gradings.
-- Yangians and W algebras.
-- equivalence between module category of W algebras and Whittaker
category of g-modules.
-- higher level Schur duality between W algebras and cyclotomic
Hecke algebras
-- (if time permits), generalizations of W algebras to the affine,
super, modular settings, etc.
Alistair Savage (University of Ottawa)
Geometric realizations of crystals
This course will follow up on and combine ideas presented in the
first week courses by Kang and Kamnitzer. In particular, Kamnitzer
will discuss various varieties used in geometric realizations of
representations of certain Lie algebras. In this course, we will
see how one can use these varieties to obtain geometric realizations
of the crystals (which are combinatorial objects) associated to
these representations and introduced in the course by Kang. A typical
construction will see the vertex set of a crystal appearing as the
set of irreducible components of certain varieties, with the crystal
operators being realized as natural geometric operations.
Vyjayanthi Chari (UC Riverside)
Representation theory of affine and toroidal Lie algebras
The plan of lectures is as follows.
Lecture 1: Representations of finite--dimensional simple Lie algerbas,
the Harishchandra homomorphism, central characters.
Lecture 2. Positive level representations of affine Lie algebras.
Lectures 3 and 4. Level zero representations of affine Lie algebras.
Lecture 5. The higher dimensional case.
June
15-27, 2009 Summer School