Dec 10Kostya TolmachovWeight yoga and ext-dualityabstract
I will finish the proof of the Beilison-Ginzburg ext-duality using the weight yoga.
Dec 03No seminar
Nov 26Thanksgiving vacation
Nov 19Kostya TolmachovExt-duality of Beilinson and Ginzburgabstract
I will explain how the simplest case of a Koszul duality in our setting follows from Soergel's Endomorphism and Structure theorems.
Nov 12No seminar
Nov 05Gus LonerganSoergel's structure theorem II
Oct 29Gus LonerganSoergel's structure theorem
Oct 22Mitya KubrakSoergel's endomorphism theorem II
Oct 15Mitya KubrakSoergel's endomorphism theorem.
Oct 8Ryan MicklerKoszul rings and Koszul duality.abstract
We will develop an understanding of Koszul duality, and what it can tell us about category O of U(g)-mod with trivial central character.
Oct 1Kostya TolmachovKazhdan-Lusztig polynomials and IC-sheaves.abstract
I will introduce some notions needed in the BGS paper and prove the classical Kazhdan-Lusztig theorem about characters of simple g-modules.
Sept 24Kostya TolmachovD-modules, category O and perverse sheaves.abstract
I will explain how to adjust the localization theorem to get an
equivalence of a block of the BGG category O with a category of perverse
sheaves on a flag variety, which is the geometric setting for the
The localization theorem is one of the most important foundational results of
geometric representation theory. It is secretly two results in one, both being some
kind of quantized version of a classical theorem, namely:
1. Kostant’s theorem relating functions on the cotangent space of the flag variety
with functions on the nilpotent cone
2. Borel-Weil-Bott theorem on behavior of certain line bundles on the flag variety
which are respectively upgraded to:
1’. Isomorphism between global sections of the sheaf of differential operators on
the flag variety with a central reduction of the universal enveloping algebra
2’. D-affine-ness of the flag variety.
In my talk, I will ‘recall’ the classical theorems of Kostant and BWB, and use them
to prove the localization theorem. Time allowing, I will go back and shed some
light on the classical theorems, which are not so trivial and often overlooked.
Localization theorem does not count as classical for the purpose of this talk, even
though Kostya may tell you otherwise.