Categorical Kazhdan–Lusztig Theory at Roots of Unity — E8 Intelligence Research

FINDING: Kazhdan–Lusztig (KL) theory for affine Weyl groups is being reformulated as a categorical equivalence at a root of unity, linking representation theory of quantum groups at q = root of unity to coherent sheaves / D-modules on the affine flag variety. | MATH: KL polynomials P_{y,w}(q) ∈ ℤ[q]; Kazhdan–Lusztig basis {C'_w} of Hecke algebra H_q(W_aff); at q = ζ primitive ℓ-th root of unity, quantum group U_ζ(𝔤) has representation category Rep(U_ζ(𝔤)) conjecturally equivalent to a category of equivariant coherent sheaves on the nilpotent cone / affine flag variety Fl_aff = G((t))/I. Key structures: affine Weyl group W_aff = W ⋉ Λ (Λ = coroot lattice), alcove geometry, affine Hecke algebra, Soergel bimodules, cells (left/right/two-sided) partitioning W_aff. | CONNECTION: Root systems and crystallographic symmetry are the native language here — W_aff is the affine crystallographic reflection group generated by reflections in hyperplanes of the coroot lattice Λ; alcoves tile the Carta Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com