Betti spectral gluing

Given a complex reductive group G, Borel subgroup B subset of G, and topological surface S with boundary partial derivative S, we study the Betti spectral category DCoh(N)(Loc(G)(S, partial derivative S)) of coherent sheaves with nilpotent singular support on the character stack of G-local systems on S with B-reductions along partial derivative S. Modifications along the components of partial derivative S endow Dcoh(N) (LOCG(S, partial derivative S)) with commuting actions of the affine Hecke category H-G in its realization as coherent sheaves on the Steinberg stack. We prove a spectral Verlinde formula identifying the result of gluing two boundary components with the Hochschild homology of the corresponding H-G-bimodule structure. The equivalence is compatible with Wilson line operators (the action of Perf(LOCG (S)) realized by Hecke modifications at points) as well as Verlinde loop operators (the action of the center of H-G realized by Hecke modifications along closed loops). The result reduces the calculation of such Betti spectral categories to the case of disks, cylinders, pairs of pants, and the Mobius band. We also show how to impose arbitrary ramification conditions in terms of modules for the affine Hecke category. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This study focuses on complex reductive groups G, Borel subgroup B, and topological surfaces S with nilpotent singular support on coherent sheaves. The spectral Verlinde formula identifies the gluing of two boundary components with the Hochschild homology of the corresponding H-G-bimodule structure. The calculation of such Betti spectral categories is reduced to simpler cases like disks, cylinders, pairs of pants, and the Mobius band.