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.

- Algebraic theory of D-modules - Bernstein
- D-modules, Perverse Sheaves, and Representation Theory - Hotta, Takeuchi, and Tanisaki
- Crystals and D-modules - Gaitsgory and Rozenblyum

**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