Langlands Duality as Dynkin Diagram Reversal and L-Function Symmetry — E8 Intelligence Research

FINDING: Langlands duality manifests as Dynkin diagram reversal — the dual group's root lattice is the coroot lattice of the original, with Weyl group symmetry preserved but weights and coweights interchanged. | MATH: For simple Lie algebra 𝔤 with root system Φ, dual 𝔤∨ has roots Φ∨ = {α∨ = 2α/⟨α,α⟩}. Dynkin diagram automorphism reverses arrows (e.g., Bₙ ↔ Cₙ, F₄ ↔ F₄, G₂ ↔ G₂). Weyl group W(Φ) ≅ W(Φ∨) — identical Coxeter groups. L-function functional equation: Λ(s,π) = ε(s,π)·Λ(1−s,π̃) where π̃ is Langlands dual representation. | CONNECTION: Root lattice ratios — for G₂, long:short root length² = 3:1; for Bₙ, ratio 2:1; for Cₙ, ratio 2:1. These ratios echo the golden-ratio-adjacent scaling in exceptional structures (E₈ has all roots equal length, giving Coxeter number 30, related to 2π/30 = 12° — base-60 compatible). The Weyl group of E₈ has order 696,729,600 = 2¹⁴·3⁵·5²·7 — highly composite, base-60 friendly. Dynkin diagram reversal preserves the Cartan matrix transpose: Aᵢⱼ = 2⟨αᵢ,α

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com