Spectral Decomposition and Hecke Eigensheaves in Geometric Langlands — E8 Intelligence Research

FINDING: Geometric Langlands program advances via spectral decomposition theorems and Hecke eigensheaf constructions on moduli stacks of G-bundles over algebraic curves | MATH: Langlands correspondence: D-mod(Bun_G) ≅ QCoh(Loc_{G^∨}); Hecke eigensheaf condition: H_V(F) ≅ V ⊗ F for all representations V of G^∨; geometric Satake: Rep(G^∨) ≅ Perv_{G(O)}(Gr_G); spectral cover: π: X̃ → X with Galois group W (Weyl group) | CONNECTION: Root systems and Weyl groups are crystallographic — the Langlands dual G^∨ exchanges roots α ↔ coroots α^∨ with ⟨α_i, α_j^∨⟩ = Cartan matrix A_ij; for ADE types these are the crystallographic Coxeter groups with exponents m_i satisfying Σm_i = |Φ|/2 + rank; the spectral cover's ramification encodes the discriminant Δ = ∏(α_i - α_j)² whose degree relates to |W| = ∏(m_i + 1) | DEPTH: 9 | --- FINDING: Modular spectral covers parameterize Hecke eigensheaves via Hitchin integrable systems, connecting Higgs bundle moduli to quantum field theory | MATH: Hitchin fibr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com