The seminar takes place in 255 Linde from 4-5:30pm on Wednesdays January - March 2019
Week 1: Reductive algebraic groups
Bowen Yang, January 23
Basic structure theory of algebraic groups; classification of reductive groups over an algebraically closed field.
- Linear Algebraic Groups - Springer
- Linear Algebraic Groups - Borel
- Algebraic Groups - Milne
- Reductive Groups - Milne
Week 2: The Tannakian formalism
Josh Lieber, January 30
Definition of tensor categories; reconstruction theorem for representations of affine group schemes.
- Tannakian categories - Deligne and Milne
- Galois Groups and Fundamental Groups - Szamuely
Week 3: The Borel-Weil-Bott theorem
Victor Zhang, February 6
Construction of irreducible representations of reductive groups as cohomology of equivariant line bundles on flag varieties; reformulation in terms of Lie algebra cohomology for the unipotent radical.
- Homogeneous Vector Bundles, Annals of Mathematics (1957) - Bott
- Lie Algebra Cohomology and the Generalized Borel-Weil Theorem, Annals of Mathematics (1961) - Kostant
- A Very Simple Proof of Bott's Theorem, Inventiones Mathematicae (1976) - Demazure
Week 4: The Springer correspondence
Lingfei Yi, February 13
Realization of Weyl group representations in the cohomology of Springer fibers; interpretation as perverse sheaves on the unipotent locus.
- A construction of representations of Weyl groups, Inventiones Mathematicae (1978) - Springer
- A topological approach to Springer's representations, Advances in Mathematics (1980) - Kazhdan and Lusztig
- Dustin Clausen's senior thesis
Week 5: The theory of D-modules
Daxin Xu, February 20
Definition of D-modules on smooth varieties; direct and inverse image functors; Kashiwara's lemma and D-modules on singular varieties; coherent, holonomic, and lisse D-modules; base change and projection formula; singular support.
Week 6: Category O and the Kazhdan-Lusztig conjectures
Zavosh Amir-Khosravi, February 27
Definition and basic properties of category O; block decomposition and translation functors; BGG reciprocity; statement of Kazhdan-Lusztig conjectures.
- Category of g-modules, Functional Analysis and its Applications (1976) - Bernstein, Gelfand, and Gelfand
- Representations of Semisimple Lie Algebras in the BGG Category O - Humphreys
- Geometric Representation Theory - Gaitsgory
- Representations of Coxeter groups and Hecke algebras, Inventiones Mathematicae (1979) - Kazhdan and Lusztig
No seminar March 6
Week 7: Beilinson-Bernstein localization
Jize Yu, March 13
The localization equivalence and its semi-classical limit; proof of Kazhdan-Lusztig conjectures.
- Localisation de g-modules, C. R. Acad. Sc. Paris (1981) - Beilinson and Bernstein
- Gaitsgory's notes, cf. Week 6
- D-modules, faisceaux pervers et conjecture de Kazhdan-Lusztig - Riche
- Perverse sheaves on flag manifolds and Kazhdan-Lusztig polynomials - Riche