D-modules and Representation Theory Reading Seminar

The Winter 2007 reading seminar on D-modules and representation theory meets on Wednesdays at 3:00 p.m. in East Hall 3096.

We are reading through Dennis' Gaitsgory's notes on geometric representation theory, which are available at http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. If time permits, we hope to talk about some other aspects of the proof of the Kazhdan-Lustztig conjectures.

Other references include:

Joseph Bernstein's notes on D-modules, at http://www.math.uchicago.edu/~arinkin/langlands/
Beilinson, Alexandre; Bernstein, Joseph Localisation de g-modules. (French) C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15--18.
Thanks to Ezra Miller for these next references:
- Steve Kleiman's history of intersection homology.
- David Kazhdan and George Lusztig's original paper, "Representations of Coxeter groups and Hecke algebras."
- Ivan Mirkovic's review of Chriss and Ginzburg's book, "Representation theory and complex geometry."
- Dragan Milicic's article on "Algebraic D-modules and representation theory of semisimple Lie groups". This is available on his website, but for those of you at Michigan, I'm also making a copy available here, since Michigan seems to block access to the Utah ftp server.

Schedule of Speakers
January 17, 2007	Sue Sierra: An overview of Category O
January 24, 2007	David Speyer: Chevalley and Harish-Chandra isomorphisms
January 31, 2007	Tatyana Chmutova: Further properties of Category O
February 7, 2007	Review of first three lectures
February 14, 2007	Alan Stapledon: Projective objects in O
February 21, 2007	Mircea Mustata: D-modules
February 24, 2007	No meeting: spring break
March 7, 2007	Mircea Mustata: D-modules, part 2
March 14, 2007	No meeting
March 21, 2007	Kyle Hofmann: Localization theory
March 28, 2007	Loren Spice: Some algebraic geometry related to g
April 4, 2007	Renzo Cavalieri: D-bundles and integrable hierarchies: a sneak preview.
April 11, 2007	Ezra Miller: Why make representation theory geometric? The Kazhdan-Lusztig conjecture via intersection cohomology

Abstracts

Renzo Cavalieri, April 4, 2007

D-bundles and integrable hierarchies: a sneak preview.

Recently Tom Nevins and David Ben-Zvi have used the geometry of D-bundles over elliptic curves to understand a classical correspondence between solutions of integrable hierarchies (KP and CM). In May we are hosting a three day workshop on this work. (see http://www.math.lsa.umich.edu/~crenzo/workshop/bn.html for more details) In this talk we will give a galloping preview of what awaits us...

Ezra Miller, April 11, 2007

Why make representation theory geometric? The Kazhdan-Lusztig conjecture via intersection cohomology

Kazhdan and Lusztig conjectured how to decompose Verma modules in terms of their irreducible submodules. Geometric representation theory suggests that instead of decomposing g-modules algebraically, one should instead work on flag varieties, where the constructible sheaves corresponding to Verma modules decompose into those for their irreducible submodules. A rough outline of the talk is as follows:
1. How Beilinson-Bernstein equivalence generalizes Borel-Weil
2. From g-modules to constructible sheaves
3. ... to intersection cohomology
4. ... to K-L polynomials
5. ... and back to g-modules.

The talk will be self-contained; prior attendance at this reading seminar will not be a prerequisite.

Last updated November 29, 2009