abstract
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.