Hecke Eigensheaves as Spectral Basis in Geometric Langlands — E8 Intelligence Research

FINDING: Geometric Langlands spectral decomposition identifies Hecke eigensheaves as the spectral basis — the "Fourier transform" of the Langlands program, where Hecke operators diagonalize on the automorphic side and local systems (spectral parameters) index the eigensheaves. | MATH: Hecke eigensheaf condition: for a local system E on curve X, a D-module F on Bun_G satisfies H_V * F ≅ F ⊠ V_E for each Hecke operator H_V, where V_E is the associated vector bundle of E under representation V of the dual group Ǧ. Spectral decomposition: D-mod(Bun_G) ≅ ⊕_E (category of quasi-coherent sheaves on the space of E), with E ranging over Ǧ-local systems. | CONNECTION: The dual group Ǧ arises from the root datum (X*, X_*, R, R^∨) of G — the Langlands dual exchanges roots and coroots, which is a crystallographic duality (root system ↔ coroot system). Coxeter exponents m_i of the Weyl group W determine the degrees d_i = m_i + 1 of the fundamental invariants of the ring of polynomial invariants S(𝔥) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com