Math 683 SPRING 2009, List of lectures

  • 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