With D. Nadler. **Loop Spaces and Langlands Parameters**.

We apply the technique of S^1-equivariant localization to sheaves on loop
spaces in derived algebraic geometry, and obtain a fundametnal
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 spaces LX.
The main observation is that this procedure connects
categories of equivariant D-modules on flag varieties with
categories of equivariant coheernt sheaves on the Steinberg variety and its
relatives. This provides a direct connetion 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. This paper forms
the first step in a project to apply the geometric Langlands program
to the complex and real local Langlands programs, which we describe.