Betti spectral gluing

Abstract

Given a complex reductive group G, Borel subgroup B⊂G, and topological surface S with boundary ∂S, we study the “Betti spectral category” DCohN(LocG(S,∂S)) of coherent sheaves with nilpotent singular support on the character stack of G-local systems on S with B-reductions along ∂S. Modifications along the components of ∂S endow DCohN(LocG(S,∂S)) with commuting actions of the affine Hecke category HG 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 HG-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 HG 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 Möbius band. We also show how to impose arbitrary ramification conditions in terms of modules for the affine Hecke category.