3,129 research outputs found
Loop Spaces and Connections
We examine the geometry of loop spaces in derived algebraic geometry and
extend in several directions the well known connection between rotation of
loops and the de Rham differential. Our main result, a categorification of the
geometric description of cyclic homology, relates S^1-equivariant quasicoherent
sheaves on the loop space of a smooth scheme or geometric stack X in
characteristic zero with sheaves on X with flat connection, or equivalently
D_X-modules. By deducing the Hodge filtration on de Rham modules from the
formality of cochains on the circle, we are able to recover D_X-modules
precisely rather than a periodic version. More generally, we consider the
rotated Hopf fibration Omega S^3 --> Omega S^2 --> S^1, and relate Omega
S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary
connection, with curvature given by their Omega S^3-equivariance.Comment: Revised versio
Loop Spaces and Representations
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.Comment: A strengthened version of the second half of arXiv:0706.0322, with
significant new material. v2: minor revisions. v3: more minor revision
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