  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. We also show that the categorical Langlands parameters for
real groups studied by Adams-Barbasch-Vogan and Soergel arise naturally from
the study of loop spaces of flag varieties and their Jordan decomposition (or
in an alternative formulation, from the study of local systems on a Moebius
strip). This provides a unifying framework that overcomes a discomforting
aspect of the traditional approach to the Langlands parameters, namely their
evidently strange behavior with respect to changes in infinitesimal character.
