Geometric Representation Theory
736 Evans Hall, Thursdays 2:10–3:30


Schedule of the talks
  1. September 3: Introduction and plan of the seminar
  2. September 10: David Kazhdan, The Satake isomorphism for Kac-Moody groups.
  3. September 17: Dario Beraldo, D-modules, constructible sheaves and the Riemann-Hilbert theorem. [1, 5, 9]
  4. September 24: Damien Mondragon, Constructible and Perverse sheaves, Hodge modules(?). [1.vii,viii,xiii, 3.viii]
  5. October 1: Harold Williams, The category O, Verma modules, central characters. [2]
  6. October 8:  George Melvin, Beilinson-Bernstein localization (for integrally-dominant regular weights). [1.xi,  6, 7, 8?, 9?]
  7. October 15: David Hill, The Hecke algebra (via Hodge modules?). [1.xiii, also possiblly 3.vii,viii]
  8. October 22: George Melvin, Kazhdan-Lusztig conjectures. [1.xii, 5.viii, 10]
  9. October 29: William Slofstra, Geometry of the Loop Grassmannian LG/G. [4, 12]
  10. November 5: Reimundo Heluani, Detailed statement of the geometric Satake correspondence. [12, 13]
  11. November 12: William Slofstra, Structure on GL-representations and its counterpart on the loop Grassmannian. [12, 13, 14, 15]
  12. November 19: Dario Beraldo, Beilinson-Bernstein equivalences for loop groups [16]
  13. December3: Constantin, Mirkovic-Vilonen cycles and T-weight spaces. Idea of proof of geometric Satake.[13]
Comments: Thanks to all the volunteers and participants!

References:
  1. Hotta, Takeuchi, Tanisaki: D-modules, Perverse Sheaves and Representation Theory, Prog. Math. 236, Birkhäuser 2008
  2. Humpreys: Representations of semisimple Lie algebras in the BGG category O; GSM 94, AMS 2008
  3. Chriss, Ginzburg: Representation Theory and Complex Geometry, Birkhäuser 1997
  4. Pressley, Segal: Loop Groups. Oxford
  5. Kirwan: An introduction to Intersection homology theory. Longman Scientific, 1988
  6. Kashiwara: Representation theory and D-modules on flag varieties. In: Orbites unipotentes et représentations, III. Astérisque  173-174
  7. Beilinson, Bernstein: Localization of g-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981)
  8. Beilinson, Bernstein: A generalization of Casselman's submodule theorem. In: Representation theory of reductive groups (Park City, Utah, 1982), 35--52, Progr. Math. 40, Birkhäuser
  9. Beilinson, Bernstein: A proof of Jantzen conjectures. I. M. Gelʹfand Seminar, 1--50, Adv. Soviet Math. 16
  10.  Brylinski, Kashiwara: Kazhdan–Lusztig conjecture and holonomic systems. Invent. Math. 64 (1981)
  11.  Hotta: Equivariant D-modules
  12. Ginzburg:  Perverse sheaves on a Loop group and Langlands' duality
  13. Mirkovic-Vilonen: Geometric Langlands duality and representations of algebraic groups over commutative rings
  14. Brylinski, Ranee Kathryn: Limits of weight spaces, Lusztig's q-analogs, and fiberings of adjoint orbits. J. Amer. Math. Soc. 2 (1989)
  15. Joseph, Letzter, Zelikson: On the Brylinski-Kostant filtration. J. Amer. Math. Soc. 13 (2000)
  16. Frenkel-Gaitsgory: D-modules on the affine flag variety and representations of affine Kac-Moody algebras
Book [1] is the basic reference for Lectures 1–8. It is on a 1-week reserve in the Library. Book [3] has among others a very different construction of the affine Hecke algebra; I don't understand the relation to [1]. I did not yet find good reference for Lecture 9.