## Graduate Seminar in Geometric Representation theory. Fall 2011

The seminar will focus on the study of the papers of Kazhdan and Lusztig on the relation between representations of affine Lie algebras

and modules over quantum groups.

The main reference is the original papers in JAMS: 1993 no. 4 (the first two papers), 1994 no. 2 (the third and fourth paper).
### Announcements

The last talk this semester will be on Dec. 6.

The notes by D.G. from the talk on the local vs global definition of fusion have been posted.

The notes by D.G. on crystals and Sugawara have been updated and now include the material from the talks on Nov. 1, 8, 15 and 22.
### Schedule

The seminar will meet on Tuesdays 5.30-8pm for a two-hour lecture (with a pizza break in the middle), alternating between Harvard and MIT,

and on Thursdays
at 5.30pm for a discussion/Q&A session in Prof. Kazhdan's office, SC 338.

Lecture 1: Sept. 13, 5.30pm, Harvard SC 507. David Kazhdan. Overview of the theory.

Lecture 2: Sept. 20. 5.30pm. MIT, room 2-190. Sasha Tsambalyuk. Strcuture of Category O.

Lecture 3: Sept. 27. 5.30pm. Harvard SC 507. Structure of Category O, continuation.

Lecture 4: Oct. 4. 5.30pm. MIT, room 2-190. Giorgia Fortuna. Tensor structure on Category O.

Lecture 5: Oct. 11. 5.30pm. Harvard SC 507. Andrei Negut. Chiral algebras and modules.

Lecture 6: Oct. 18. 5.30pm. MIT, room 2-190. Andrei Negut. The Sugawara construction.

No seminar on Oct. 25.

Lecture 7: Nov. 1. 5.30pm. MIT, room 2-190. Dennis Gaitsgory. Sugawara and universal constructions (a rerun of Andrei's talk).

Lecture 8: Nov 8. 5.30pm. Harvard, room 507. Dennis Gaitsgory. Continuation of the previous talk.

Lecture 9: Nov. 15. 5.30pm. MIT, room 2-190. Dennis Gaitsgory. Structure of crystal on the category of chiral modules.

Lecture 10: Nov. 22. 5.30pm. Harvard, room 507. Dennis Gaitsgory. Finally, the Sugawara construction.

Lecture 11: Nov. 29. 5.30pm. MIT, room 2-190. Dennis Gaitsgory. Local definition of the KL tensor product.

Lecture 12: Dec. 6. 5.30pm. Harvard, room 507. Dennis Gaitsgory. Recovering the full KL tensor product.
### Seminar Notes

- Sept. 20-27,
Category O at the negative level, by Sasha Tsymbalyuk.
- Oct. 4,
Monoidal structure on category O, by Giorgia Fortuna.
- Oct. 11, Connecting coinvariants, by Andrei Negut.
- Nov. 1, 8, 15 and 22, Universal constructions of crystals and Sugawara, by Dennis Gaitsgory.
- Nov. 29, Llocal vs global definition of the fusion tensor product, by Dennis Gaitsgory.