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Geometric rep theory

Jan 2022

3 min read

Our goal is to understand the basic tools of geometric representation theory. Geometric representation theory attempts to understand symmetry objects like groups, Lie algebras, quantum groups, etc via their actions on objects of a geometric nature – which are expected to be more fundamental than representation-theoretic data. As an example of the power of these techniques, consider the independent geometric proofs of the Kazhdan-Lusztig conjectures by Beilinson-Bernstein and Brylinski-Kashiwara.

This semester, we will attempt to understand at least some of the following topics:

Some references are:

Schedule

DateSpeakerTitleAbstract
Feb 02—Organizational meeting
Feb 09Fan ZhouD-modules, bare minimumz, part 1We discuss basics on D-modules. We plan to get to pushforwards and pullbacks and Kashiwara’s theorem, with maybe some discussions of consequences. No representation theory is happening this talk sadly.
Feb 16Fan ZhouD-modules, bare minimumz, part 2We continue on D-module basics. We’ll try to give more examples this time to build intuition for how these things look and behave like. Having spent too long on right-left transfer modules last time, hopefully (we had better!) get to pushforwards/pullbacks and Kashiwara.
Feb 23—Seminar cancelled
Mar 02Fan ZhouD-modules, bare minimumz, part 3We finish preliminaries on D-modules by talking about consequences of Kashiwara, the derived setting, and the de Rham complex. Next time we’ll either give a crash course on algebraic groups or continue beyond bare minimums in D-modules.
Mar 09Fan Zhoubrief interlude: refresher on algebraic groupsWe give a speedrun through the main dictionary/features of algebraic groups. We will give no proofs – this is strictly a survey. The point is to familiarize ourselves with the players on the field for our next talk, when we finally begin representation theory.
Mar 23Fan Zhoulocalization: d-modules and representationzWe finally begin localization proper. We will begin by rushing through the necessary constructions and stating the localization theorem. We’ll also try to say something about Borel-Weil-Bott.
Mar 30Fan Zhouproof of localization: d-modules and representationz, part 1We begin by computing an example of localization. Then we will sketch the proof of localization and begin filling in the details. Hopefully this will take two talks.
Apr 06Fan Zhouproof of localization: d-modules and representationz, part 2We finish the proof of localization. Recall what was left was some representation theory and some algebraic geometry. While the representation theory outlook is not the standard, we will do it anyway. Through the rep theory we will confirm that our statement of localization has the correct twist.
Apr 13Fan Zhousome more on D-modules and identifying some highest-weight modules in the setupPreviously, we only did the bare minimum on D-modules to prove localization. Now we pay some old debts and say something more about D-modules.
Apr 20Kevin ChangThe Riemann-Hilbert correspondenceI’ll review constructible sheaves and then say a bit about the Riemann-Hilbert correspondence.
Apr 27Kevin ChangThe Kazhdan-Lusztig conjecturesIn Fan’s most recent talk, we discussed the images of Vermas and simples under Beilinson-Bernstein localization. The Riemann-Hilbert correspondence turns these D-modules into perverse sheaves. I’ll explain how this leads to a proof of the Kazhdan-Lusztig conjectures by relating the Hecke algebra to the geometry of Schubert varieties.
May 11Patrick LeiWhat’s the deal with Nakajima quiver varieties?I will introduce Nakajima quiver varieties and attempt to explain why so many people care about them. Notes