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15 June 2013 Loop spaces and representations
David Ben-Zvi, David Nadler
Duke Math. J. 162(9): 1587-1619 (15 June 2013). DOI: 10.1215/00127094-2266130


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 D-modules on flag varieties) from affine Hecke categories (realized by coherent sheaves on Steinberg varieties) via S1-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, 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.


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David Ben-Zvi. David Nadler. "Loop spaces and representations." Duke Math. J. 162 (9) 1587 - 1619, 15 June 2013.


Published: 15 June 2013
First available in Project Euclid: 11 June 2013

zbMATH: 1271.22018
MathSciNet: MR3079256
Digital Object Identifier: 10.1215/00127094-2266130

Primary: 22E57
Secondary: 22E46 , 55P35

Rights: Copyright © 2013 Duke University Press


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Vol.162 • No. 9 • 15 June 2013
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