Categorical Geometric Langlands via Ind-Coherent Sheaves and ∞-Categories — E8 Intelligence Research

FINDING: Categorical geometric Langlands duality is formulated via ind-coherent sheaves on stacks, requiring ∞-category theory to capture the full symmetry; the duality exchanges automorphic objects (D-modules) with spectral objects (ind-coherent sheaves) under a Fourier–Mukai-type transform. | MATH: The central equivalence is \( \text{IndCoh}(\text{LocSys}_{G^\vee}) \simeq \text{D-mod}(\text{Bun}_G) \) (categorical Langlands), where \(G^\vee\) is the Langlands dual group. Key structures: ∞-categories, ind-coherent sheaves (vs. quasi-coherent), Springer correspondence as a categorical limit, and the "coherent Springer theory" linking nilpotent cones to Weyl group representations. No explicit numerical constants appear; the deep content is structural (functoriality, dualities of stacks). | CONNECTION: The Langlands dual group \(G^\vee\) encodes root system duality — e.g., for \(G = \mathrm{SO}(2n+1)\), \(G^\vee = \mathrm{Sp}(2n)\), exchanging long/short roots. This is a crystallographic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com