Introduction to Geometric Representation Theory

These notes are a rough write-up of lectures presented at the workshop on “Geometry and Integrability”, held at the University of Melbourne, 6-15 Feb, 2008. My interest in Geometric Representation Theory stems from recent developments in the Geometric Langlands program. These lectures are an introduction into material which are a prerequisite to anyone interested in studying the Geometric Langlands program. Most of the results I will discuss can be found in the textbooks by Pressley and Segal [PS], Chriss and Ginzburg [CG], and Hotta, Takeuchi and Tanisaki [HTT], or the lecture notes by Bernstein [Ber] and Gaitsgory [Gai]. Here one can also find references to the original papers. My presentation of the material was very much inspired by, and to some extent based on, lectures given by Ian Grojnowski at the “Introduction to the Geometric Langlands Program” [Gro]. The lectures were aimed at Mathematical Physicists with a similar background as myself. While I have tried to be reasonably precise, I have occasionally compromised and chosen to convey an idea, often by example, rather than spelling out all the mathematical details, in particular those that require more advanced knowledge of algebraic geometry. Needless to say that in three one-hour lectures one can only scratch the surface of the subject. My hope is that these lectures at least convey the beauty of Geometric Representation Theory and that they may motivate the reader to further study.