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Math 683 SPRING 2009, List of lectures
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On this page I will post content of all lectures. All handouts also will be posted here.
Monday, March 30: Overview.
Wednesday, April 1: Chevalley basis.
Friday, April 3: Chevalley groups of adjoint type.
Monday, April 6: Definition of category O and first properties.
Wednesday, April 8: Category O is noetherian.
Friday, April 10: Category O is artinian. Simple objects in category O.
Hom's are finite dimensional in category O.
Monday, April 13: Action of the center of U(g) on the category O.
Wednesday, April 15: Ext^1 and decomposition into blocks.
Friday, April 17: More on blocks. Group W_[\lambda].
Some Ext^1-calculations.
Monday, April 20: Duality in category O.
Wednesday, April 22: Duals of Verma modules.
Dominant and anti-dominant weights.
Friday, April 24: Projective objects in category O.
Monday, April 27: Standard filtrations.
Wednesday, April 29: Indecomposable projectives.
Friday, May 1: BGG reciprocity.
Wednesday, May 6: Indecomposable objects in category O of sl(2).
Friday, May 8: Every Verma module has a unique simple submodule.
Monday, May 11: Embeddings of Verma modules, I.
Wednesday, May 13: Embeddings of Verma modules, II.
Friday, May 15: Simple Verma modules and blocks in category O.
Monday, May 18: Strong linkage and BGG theorem. Bruhat ordering.
Please find FINAL EXAM here ; it is due on
Thursday, June 11, noon (Deady 10B).
Wednesday, May 20: Jantzen filtration and Jantzen sum formula.
Friday, May 22: Contravariant form and Shapovalov formula.
Wednesday, May 27: Shapovalov formula and Jantzen filtration.
Friday, May 29: Complements: BGG resolution, Ext's calculations,
homological dimension of category O. Induced representations of finite groups
and Hecke algebras.
Monday, June 1: Back to groups over finite fields: inducing from
Borel subgroup and Iwahori-Hecke algebra.
Wednesday, June 3: Kazhdan-Lusztig basis of Iwahori-Hecke algebra.
Kazhdan-Lusztig conjecture.
Friday, June 5: Outline of the proof of Kazhdan-Lusztig conjecture.
END