Introduction to Geometric Representation Theory
Student No.:40
Time:Mon & Wed 15:20-16:55, Sep.17-Dec.12
Instructor:单芃Shan Peng  
Place:Conference Room 1, Jin Chun Yuan West Bldg.
Starting Date:2018-9-17
Ending Date:2018-12-12

In this course, we will explain how to use geometric objects, such as D-modules and perverse sheaves on flag varieties, to study representations of semi-simple Lie algebras.

In 1979, a deep link between representations of semi-simple Lie algebras over a field of characteristic zero and perverse sheaves on flag varieties was discovered by Kazhdan-Lusztig. In particular, they proposed some conjectural formulae for characters of simple highest weight representations in terms of intersection cohomology of Schubert varieties. This conjecture was proved soon after by Beilinson-Bernstein and Brylinski-Kashiwara by localizing representations of Lie algebras to D-modules on flag varieties. This theory has now become a fundamental result in geometric representation theory and has seen a lot of developments. It will be the main focus of this course.

If time permits, we will also explain a beautiful result of Bezrukavnikov-Mirkovic-Rumynin relating modular representations of semi-simple Lie algebras to coherent sheaves on the cotangent bundle of flag varieties.

Some basic knowledge on representations of semi-simple Lie algebras and on algebraic geometry would be useful, but not indispensable.

[1] Hotta, R., Kiyoshi Takeuchi, and Toshiyuki Tanisaki. D-Modules, Perverse Sheaves, and Representation Theory. English ed. Progress in Mathematics, v. 236. Boston: Birkhäuser, 2008;
[2] Humphreys, J. Representations of semisimple Lie algebras in the BGG category 𝒪. Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI, 2008;
[3] Bezrukavnikov, R., Mirković, I., Rumynin, D., Localization of modules for a semisimple Lie algebra in prime characteristic. With an appendix by Bezrukavnikov and Simon Riche. Ann. of Math. (2) 167 (2008), no. 3, 945–991.