Program
On the category O of semisimple Lie algebras
Student No.:50
Time:Wed/Fri 13:00-14:50, 2014-07-30 ~ 2014-08-22 2014-09-10 ~ 2014-09-24
Instructor:Peng Shan  [CNRS - Université de Caen]
Place:Conference Room 3, Floor 2, Jin Chun Yuan West Building
Starting Date:2014-7-30
Ending Date:2014-9-24
 

Description:

 

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.

 
 

Prerequisite:

 

Familarities with semisimple Lie algebras, flag varieties and basic sheaf theories would be helpful but not indispensable.

 
 

Reference:

 

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.