Winter 23/24 - Representation Theory II - Spaces In Geometric Representation Theory

We study the geometry of some spaces that play an important role in the (geometric) representation theory of complex reductive groups and Lie algebras.
The course starts with a refresher on the classification, structure and representation theory of complex reductive algebraic groups. We then discuss the geometry of flag varieties and related spaces, with a special attention to the singularities of Schubert varieties. We will see how these notions generalise to Kac-Moody groups and study affine Grassmannians.
Moreover, we discuss conjugacy classes in reductive groups and the Grothendieck—Springer resolution.
If time permits, a crash course on quotient stacks will allow us to discuss the horocycle correspondence.

• Algebra I & II
• knowledge of representation theory of groups and Lie algebras, as well as structure theory of semisimple Lie-algebras/groups (familiarity with representation theory is expected, not just the classification theorems)
• basic knowledge in algebraic geometry, for example, of projective complex varieties

• Classification des groupes de Lie algébriques. Séminaire Claude Chevalley 1956-1958. Tomes 1, 2
• Schémas En Groupes (SGA3)
• Linear Algebraic Groups, Humphreys, James E.
• Linear Algebraic Groups, Borel, Armand
• Introduction to Actions of Algebraic Groups, Brion, Michel
• Representations of Algebraic Groups, Jantzen, Jens Carsten
• ... more to come :)

Lectures

Wednesday 10-12, Room: Großer Hörsaal,
Friday 12-14, Room: Zeichensaal.

Problem session

Wednesday 14-16, Room: 0.006.

There will be exercise problems posed in the lectures.

Everyone is encouraged to present their solutions on the blackboard during the problem session. You should do this at least once.

Exams

There will be an oral exam at the end of the semester. More information on the dates will follow.

Links

The BASIS entry for the lecture and problem session.