A spectral incarnation of affine character sheaves. (30th June 2017)

• Record Type: Journal Article
• Authors: Ben-Zvi, David; Nadler, David; Preygel, Anatoly
• Is Part Of: Compositio mathematica. Volume 153:Number 9(2017)
• Page Start: 1908
• Page End: 1944
• Publication Date: 2017-06-30
• Subjects: 20C08, -- 14D24 (primary); geometric Langlands program, -- affine Hecke algebra, -- character sheaves; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007278
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions, which appear in recent advances in the geometric Langlands program. The key technical tools in our arguments are a new descent theory for coherent sheaves or ${\mathcal{D}}$ -modules with prescribed singular support and the theory of integral transforms for coherent sheaves developed in the companion paper by Ben-Zvi et al . [ Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS), to appear].