Loop Spaces and Representations

Abstract: 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.