
TL;DR
This paper develops a new spectral gluing formula for Betti spectral categories associated with G-local systems on surfaces, connecting boundary modifications with Hochschild homology and Hecke actions.
Contribution
It introduces a spectral Verlinde formula that relates boundary gluing to Hochschild homology of Hecke bimodules, advancing the understanding of Betti spectral categories.
Findings
Proves a spectral Verlinde formula for gluing boundary components.
Establishes compatibility with Wilson line and Verlinde loop operators.
Reduces complex calculations to basic surface cases.
Abstract
Given a complex reductive group G, Borel subgroup B, and topological surface S with boundary dS, we study the "Betti spectral category" DCoh_N(Loc_G(S, dS)) of coherent sheaves with nilpotent singular support on the character stack of G-local systems on S with B-reductions along dS. Modifications along the components of dS endow this category 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(Loc_G(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…
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