Weyl Groups, Springer Theory, and Equivariant Khovanov Homology — E8 Intelligence Research

FINDING: The intersection of Weyl group actions, Springer theory, and equivariant Khovanov homology reveals a deep structural correspondence between Lie-theoretic geometry (nilpotent orbits, Springer fibers) and categorified quantum invariants, with explicit symmetries (involutions, integral lifts) emerging in equivariant settings. MATH: - **Springer correspondence**: Irreducible representations of Weyl group \\(W\\) ↔ pairs (nilpotent orbit \\(\\mathcal{O}\\), local system on \\(\\mathcal{O}\\)). Generalized Springer fibers \\(S_\\mu\\) (for nilpotent \\(\\mu\\)) admit pavings by affine spaces (Graham). - **Khovanov homology**: Categorifies Jones polynomial; odd Khovanov homology uses arc algebras \\(A_n\\) with \\(\\mathbb{Z}/2\\)-grading; real Springer fibers \\(\\mathcal{R}_\\lambda\\) appear as geometric models for these algebras (Wilbert). - **Equivariant symmetries**: For \\(U(2)\\)-equivariant Khovanov homology, an involution \\(\\widehat{\\sigma}\\) exists; an integral lift \\(\\widehat{\\nu}\\) of Sh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com