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 IndCoh₍₈₋₏ (Par ₆) by a coherent Springer sheaf SG, 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 IndCoh (Comm ₆). The paper includes (i) a detailed worked construction for type A (notably G=GLₙ (Fq), q) and (ii) a first non-type-A case study G=Sp₄ (Fq) at 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.