Langlands Duality Induces Rational K-Theory Isomorphism for Affine Weyl Groups — E8 Intelligence Research

FINDING: Langlands duality induces a rational isomorphism in K-theory between group C*-algebras of extended affine Weyl groups, via equivariant Poincaré duality on dual tori. MATH: Let \( \check{T} \) and \( T \) be dual tori (Langlands dual), with finite group actions \( W \) (Weyl group). Construct equivariant Poincaré duality: \( K^*_W(\check{T}) \cong K^{\dim T - *}_W(T) \) rationally. Extended affine Weyl group \( \tilde{W} = W \ltimes Q^\vee \) (coroot lattice). Then \( K_* (C^*(\tilde{W})) \otimes \mathbb{Q} \cong K_* (C^*(\tilde{W}^\vee)) \otimes \mathbb{Q} \), where \( \tilde{W}^\vee = W \ltimes P \) (weight lattice). Key constants: rank \( n \), torsion primes dividing \( |W| \) inverted. No explicit numeric ratios appear — the structure is lattice-theoretic. CONNECTION: Direct link to root systems (crystallographic symmetry). The dual tori \( T, \check{T} \) encode weight vs. coroot lattices — the ratio of their covolumes is the index \( |P/Q^\vee| \) (fundamental group Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com