Junior Geometry Seminar – Chingun Erdenebileg

Abstract: D-modules are modules over the sheaf of differential operators, a non-commutative expansion of the structure sheaf of a variety. They, surprisingly, are one of the key tools in geometric representation theory – the representation theory of Lie algebras can be realised via D-modules on the associated flag variety via Beilinson-Bernstein localization, which spurred on the proof of the Kazhdan-Lusztig conjecture.

In this talk, I’ll give an overview of their basic definitions and theory, leading up the derived equivalence of certain D-modules with perverse sheaves and the result of Beilinson-Bernstein, then briefly discuss their G-equivariant version if time permits.