Langlands Correspondence and Bezrukavnikov's equivalence, part 2
Typed notes by Anna Romanova (usually up to date within a week or so) are here:
https://www.maths.usyd.edu.au/u/romanova/Syllabus/LanglandsandBezrukavnikov.pdf
Lecture 1: Review of LLC, unramified correspondence, tamely ramified correspondence. Affine and extended affine Weyl group.
Lecture 2: Iwahori-Matsumoto Hecke algebra, Bernstein presentation, centre, central characters and Deligne-Langlands conjecture.
Informal Friday seminar lecture 1: Geometry of the affine Grassmannian.
Lecture 3: Springer resolution, examples of Springer fibres, conormal space, Steinberg variety.
Informal Friday seminar lecture 2: The geometric Satake equivalence.
Lecture 4: Springer correspondence, Borel-Moore homology, convolution algebras, connection to Springer corresponndence.
Lecture 5: More on convolution algebras, Kazhdan-Lusztig equivalence, easy consequences, Bezrukavnikov's equivalence (rough statement)
Pictures of the cells I discuss briefly are here.
NOTE: NO LECTURE ON 25/10!
Lecture 6: (6/3) Constructible and perverse sheaves on curves.
Lecture 7: (13/3) Nearby cycles, vanishing cycles and Beilinson glueing on curves.
NOTE: NO LECTURE ON 20/3!
Lecture 8: (27/3) (online) Overview of proof that derived category of perverse sheaves agrees with constructible derived category.
Lecture 9: (3/4) (online) Highest weight categories. Derived category of perverse sheaves with respect to a stratification by affine spaces agrees with constructible derived category. Back to Kazhdan-Lusztig isomorphism.
Lecture 10: (10/4) (online) Equivariant K-theory of Steinberg again, anti-spherical module, start of outline of proof of Kazhdan-Lusztig isomorphism.
Lecture 11: (17/4) (online) Completion of proof of Kazhdan-Lusztig isomorphism via action on anti-spherical module.
Lecture 12: (24/4) (online) Rough statement of Arkhipov-Bezrukavnikov. Gaitsgory's central sheaves, example of natural representation of GL_2.
Exercise sheet: Bernstein centre for affine Hecke algebra of GL_2.
Lecture 13: (1/5) (online) A hitchhiker's guide to the Hecke category.
Lecture 14: (8/5) (online) The categorical anti-spherical category and its symmetries.
Lecture 15: (15/5) (online) Monoidal categories and their modules. Strict group actions. Representations of representations of SL_2.
Lecture 16: (22/5) (online) Modules over algebras in monoidal categories. Abelian categories over stacks.
Lecture 17: (29/5) (online) (De)equivariantisation.
Lecture 18: (5/6) (online) Coherent sheaves on projective space and base affine space.
Lecture 19: (12/6) (online) Constructible sheaves on flag varieties and braid groups.
Lecture 20: (19/6) (online) Affine flags, affine braids and Wakimoto sheaves.
Lecture 21: (26/6) (online) Whittaker functions, Whittaker sheaves and the Arkhipov, Bezrukavnikov theorem.
Lecture 22: (3/7) Soergel bimodules, Soergel calculus and BGK central complexes, following Elias.
Lecture 23: (10/7) Potted history of Koszul duality, torus monodromic sheaves Ă la Bezrukavnikov-Yun, statement of Bezrukavnikov's equivalence.