The goal of the course is give a systematic introduction to the category O of a semi simple Lie algebra, which is a central object in representation theory. We expect to cover the following topics:
1. Highest weight structure of the category O.
2. The Kazhdan-Lusztig conjecture.
3. Beilinson-Berstein’s localization theorem, which relates the category O to D-modules on flag varieties, and yields a (geometric) proof of the KL-conjecture.
4. Soergel’s bimodules and an algebraic proof of the KL-conjecture by Elias-Williamson.
5. Relation with categorifications of Hecke algebras and Kac-Moody algebras.
6. Perspectives in positive characteristics.
Familarities with semisimple Lie algebras, flag varieties and basic sheaf theories would be helpful but not indispensable.
J. Humphreys, Representations of Semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, Vol. 94, AMS.
R. Hotta,K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory,Progress in Mathematics, 236. Birkhäuser Boston, Inc., Boston, MA, 2008.
B. Elias, G. Williamson, The Hodge theory of Soergel bimodules, arxiv:1212.0791, to appear in Annals of Math.