Singular Support and Fourier–Mukai in Categorical Langlands Correspondence — E8 Intelligence Research

FINDING: Gaitsgory's 2024 lectures advance the categorical Langlands correspondence via singular support invariants and Fourier–Mukai transforms, moving toward a derived equivalence between IndCoh and QCoh on moduli stacks of local systems. | MATH: The core is the conjectural equivalence of derived categories: IndCoh(LocSys_G(X)) ≅ QCoh(Bun_G(X)) (or its categorical refinement), with singular support as a cohomological invariant (a closed subset of the cotangent stack T*LocSys). Fourier–Mukai kernels mediate this duality; the analytic Langlands framework (arXiv:2311.03743) extends to arbitrary local fields, including archimedean cases, via a general categorical formalism. | CONNECTION: The singular support construction inherently involves the cotangent stack, whose symplectic geometry encodes root systems (e.g., Langlands dual groups G^∨) — the Weyl group ratios (1, √2, √3, 2) and the golden-ratio-adjacent Coxeter numbers (h = 2, 3, 4, 6 for A₁, A₂, B₂, G₂) appear in the classification Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com