Unipotent Elements and Twisting in Link Homology

Abstract

Let $\mathcal{U}$ be the unipotent variety of a complex reductive group $G$. Fix opposed Borel subgroups $B_{\pm }\subseteq G$ with unipotent radicals $U_{\pm }$. The map that sends $x_{+}{x}_{-}\mapsto x_{+}{x}_{-}{x}_{+}^{-1}$ for all $x_{\pm }\in U_{\pm }$ restricts to a map from $U_{+}{U}_{-}\cap g{B}_{+}$ into $\mathcal{U}\cap g{B}_{+}$, for any $g$. We conjecture that the restricted map forms half of a homotopy equivalence between these varieties, and thus, induces a weight-preserving isomorphism between their compactly-supported cohomologies. Noting that the map is equivariant with respect to certain actions of $B_{+}\cap g{B}_{+}{g}^{-1}$, we prove for type $A$ that an equivariant analogue of this isomorphism exists. Curiously, this follows from a certain duality in Khovanov-Rozansky homology, a tool from knot theory.

18 pages