In the second part of the course, in the Spring, we will study D-modules on general
algebraic varieties.
This would require from the beginning to use language of sheaves and use more
sophisticated homological technique (derived categories). We will spend some time
discussing these subjects.
Of course at this stage we will have to assume that people know some basic properties
of coherent sheaves on algebraic varieties (or at least have heard about them).
In other words, we will try to make my lectures formally self contained, but a prior
knowledge of basic facts of algebraic geometry would help.
We are also planning to try to describe how the language of D-module theory allows
to give a geometric interpretation of representations of reductive Lie groups and Lie
algebras (localization of g-modules).
This localization technique is a very powerful tool that allows to prove many highly non-trivial results in representation theory. It also connects Representation Theory with other areas of Mathematics (D-modules. perverse sheaves, weight filtrations and so on).
If time permits we will discuss some of these subjects in more details.