Modular Coherent Springer Theory and Categorical Langlands for Finite Groups

This preprint develops a “coherent Springer → categorical Langlands” bridge for finite groups of Lie type, organized around a three-axiom Langlands generator package (LG1–LG3): compact generation of \mathrm{IndCoh}_{\mathrm{nilp}}(\mathrm{Par}_{\check G}) by a coherent Springer sheaf S_G, identification of its endomorphism algebra with the group algebra (in the rather-good range), and compatibility of the diagonal coalgebra structure with Hopf comultiplication. The framework yields monoidal integral-transform equivalences and corresponding identifications after taking Drinfeld centers, relating D(kG)-modules to \mathrm{IndCoh}(\mathrm{Comm}_{\check G}). The paper includes (i) a detailed worked construction for type A (notably G=\mathrm{GL}_n(\mathbb{F}_q), \ell\nmid q) and (ii) a first non-type-A case study G=\mathrm{Sp}_4(\mathbb{F}_q) at \ell\ge 3, proving a t-exact spectral identification that intertwines the block decompositions (7 simples across 2 blocks) via both a classical assembly (AHJR, BZCHN, Rider–Russell, BFO) and an independent route through Kato’s exotic nilpotent cone. The general-theory statements are written with an explicit dependency diagram: outside the rather-good range, only the modular endomorphism identification (LG2) is treated as conditional, while the generator (LG1) and Hopf-diagonal compatibility (LG3) are formulated as unconditional in the paper’s logical order.