Langlands Duality Induces Rational K-Theory Isomorphism for Affine Weyl Groups — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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 checkT and T be dual tori (Langlands dual), with finite group actions W (Weyl group). Construct equivariant Poincaré duality: K*W(checkT) ≅ Kdim T − *W(T) rationally. Extended affine Weyl group W̃ = W ltimes Q∨ (coroot lattice). Then K* (C*(W̃)) ⊗ ℚ ≅ K* (C*(W̃∨)) ⊗ ℚ, where W̃∨ = 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, checkT encode weight vs. coroot lattices — the ratio of their covolumes is the index |P/Q∨| (fundamental group
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Phi-Harmonic E8 Lattice Mirrors Langlands Categorical Duality — E8 Intelligence Research
- Quantum Geometric Langlands Duality Linked to Symplectic Duality and Affine Weyl Structure — E8 Intelligence Research
- Geometric Langlands Conjecture Proved: Galois-Automorphic Duality via Sheaves — E8 Intelligence Research
- Quantum Geometric Langlands: A Categorical Equivalence via Fundamental Local Duality — E8 Intelligence Research
- Langlands Duality as Dynkin Diagram Reversal and L-Function Symmetry — E8 Intelligence Research
BibTeX
@misc{caldin2026langlands,
title = {{Langlands Duality Induces Rational K-Theory Isomorphism for Affine Weyl Groups — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658273},
url = {https://zenodo.org/records/23658273},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Langlands Duality Induces Rational K-Theory Isomorphism for Affine Weyl Groups — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658273
UR - https://zenodo.org/records/23658273
AB - 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 checkT and T be dual tori (Langlands dual), with finite group actions W (Weyl group). Construct equivariant Poincaré duality: K^*_W(checkT) ≅ K^dim T − *_W(T) rationally. Extended affine Weyl group W̃ = W ltimes Q^∨ (coroot lattice). Then K_* (C^*(W̃)) ⊗ ℚ ≅ K_* (C^*(W̃^∨)) ⊗ ℚ, where W̃^∨ = 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, checkT encode weight vs. coroot lattices — the ratio of their covolumes is the index |P/Q^∨| (fundamental group Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -