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 our previously developed theory to flag varieties, and we obtain new insights into fundamental categories in representation theory. First, we show that one can recover finite Hecke categories (realized by $\mathcal{D}$-modules on flag varieties) from affine Hecke categories (realized by coherent sheaves on Steinberg varieties) via $S^1$-equivariant localization. Similarly, one can recover $\mathcal{D}$-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, and Vogan and by 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 Möbius 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.