Unipotent Elements and Twisting in Link Homology

TL;DR

This paper explores a conjectured homotopy equivalence involving unipotent varieties in complex reductive groups and links it to Khovanov-Rozansky homology, revealing new connections between algebraic geometry and knot theory.

Contribution

It proposes a conjecture about a homotopy equivalence between certain varieties and proves an equivariant version for type A groups, connecting it to knot homology.

Findings

Conjecture of a homotopy equivalence between unipotent-related varieties.

Proof of an equivariant isomorphism in type A groups.

Connection established between algebraic geometry and Khovanov-Rozansky 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 $U_+U_- \cap gB_+$ into $\mathcal{U} \cap gB_+$, 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 $B_+ \cap gB_+g^{-1}$, 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.