Pavle Pandzic
University of Zagreb, Croatia
D-modules and representation theory

D-modules have entered representation theory of Lie groups and algebras through the work of Beilinson and Bernstein, who proved the Jantzen conjecture and consequently also the Kazhdan-Lusztig conjecture for highest weight modules.

The main idea is that representations of a Lie algebra $\mathfrak g$ with fixed infinitesimal character can be localized to obtain a sheaf of modules over the sheaf of twisted differential operators on the flag variety of $\mathfrak g$. Sheaves are more complicated objects than modules, but they have an important advantage: one can study them in local coordinates. This makes the approach very powerful, while also quite involved and sophisticated.

In these lectures we will first study the basics of D-modules on $\mathbb C^n$, that is, the modules over the Weyl algebra. We will then pass to more general algebraic varieties, and describe the basic constructions like inverse and direct image functors.

We will then pass to the setting of flag varieties, describe the localization functor of Beilinson and Bernstein and their classification of irreducible holonomic D-modules.

Everything will be illustrated by examples, notably by the case $\mathfrak g=sl(2,\mathbb C)$. A good reference is the paper of D. Mili\v ci\'c, Algebraic D-modules and representation theory of semisimple Lie groups from Analytic Cohomology and Penrose Transform, M. Eastwood, J.A. Wolf, R. Zierau, editors, Contemporary Mathematics, Vol. 154 (1993), 133-168. (pdf file available from http://www.math.utah.edu/ftp/u/ma/milicic/math/penrose.pdf ).