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The seminar takes place in 255 Linde from 4-5:30pm on Wednesdays.

Week 1: Reductive algebraic groups

Bowen Yang, January 23

Basic structure theory of algebraic groups; classification of reductive groups over an algebraically closed field.

Week 2: The Tannakian formalism

Josh Lieber, January 30

Definition of tensor categories; reconstruction theorem for representations of affine group schemes.

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.