Loop Spaces and Langlands Parameters

Understand

We apply the technique of S^1-equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory.

• Namely, we categorify the well known relationship between free loop spaces, cyclic homology and de Rham cohomology to recover the category of D-modules on a smooth stack X as a localization of the category of S^1-equivariant coherent sheaves on its loop space LX.
• The main observation is that this procedure connects categories of equivariant D-modules on flag varieties with categories of equivariant coherent sheaves on the Steinberg variety and its relatives.
• This provides a direct connection between the geometry of finite and affine Hecke algebras and braid groups, and a uniform geometric construction of all of the categorical parameters for representations of real and complex reductive groups.