p-Kazhdan-Lusztig Polynomials Unify Modular Characters and Tilting Multiplicities — E8 Intelligence Research

FINDING: The p-Kazhdan-Lusztig polynomials govern modular character formulas for reductive groups in positive characteristic, linking perverse sheaves on affine flag varieties to tilting module multiplicities. | MATH: The key object is the p-Kazhdan-Lusztig basis \(\{ {C}_w \}\) of the affine Hecke algebra over \(Z\), specializing at \(v ↦ 1\) with \(p\)-dependent coefficients. The character formula (Williamson–Riche) states: \(ch(L(λ)) = ∑μ ≤ λ (-1)(λ)-(μ) Pμ,λ⁽ᵖ⁾(1) \, ch(Δ(μ))\), where \(Pμ,λ⁽ᵖ⁾\) are p-KL polynomials. These satisfy \(Pμ,λ⁽ᵖ⁾ = ∑ᵢ a_i v^i\) with \(a_i ∈ Z≥ 0\), and crucially \(Pμ,λ⁽ᵖ⁾ ≠ Pμ,λ\) (classical KL) for small \(p\). The affine flag variety \(G((t))/I\) (Iwahori) has perverse sheaves whose stalks compute these polynomials; the modular case requires \(F_p\)-coefficient Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com