Math 735 — Topics in Algebra:

Geometric Representation Theory

Fall 2002

Instructor: Milen Yakimov

Time: TR 11:40-12:55

Room: Malott 230

One of the most powerful methods in representation theory is the application of geometric tools which often has numerous advantages compared to a pure algebraic or cohomological treatment. On the other hand Lie theory supplies lots of examples of spaces with rich geometric structure.

We will follow very closely the book of Victor Ginzburg and Neil Chriss. Our goal is to explain some instances of this interaction without assuming any background in algebraic geometry or Lie theory. In this way the course will be accessible to all graduate students. In particular, students in representation theory and algebraic geometry can benefit a lot from a course on the interaction between these fields.

Starting from scratch we will:

1. Develop some needed geometric techniques, define and discuss properties of the functors of Borel-Moore homology and equivariant K-theory, and
    
2. Define some major geometric spaces associated to a complex simple Lie algebra as the flag variety, the nilpotent cone, the Springer variety and study them.

Then we will use the functors of Borel-Moore homology and K-theory to define and study representations of Lie algebras and Hecke algebras.

If time permits we will also introduce Nakajima's quiver varieties and discuss briefly geometric representations of Kac-Moody algebras and quantum groups. Since this is only a semester long course we will need to leave uncovered large areas of geometric representation theory, e.g., applications of D-modules.

Text: Neil Chriss and Victor Ginzburg: Representation Theory and Complex Geometry. Birkhäuser, Boston 1997.