Loop Spaces and Representations

Understand

We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups.

• In particular, we apply the theory developed in our previous paper arXiv:1002.3636 to flag varieties, and obtain new insights into fundamental categories in representation theory.
• First, we show that one can recover finite Hecke categories (realized by D-modules on flag varieties) from affine Hecke categories (realized by coherent sheaves on Steinberg varieties) via S^1-equivariant localization.
• Similarly, one can recover D-modules on the nilpotent cone from coherent sheaves on the commuting variety.