The geometry of Markov traces

TL;DR

This paper provides a geometric interpretation of the Jones-Ocneanu trace using equivariant cohomology, generalizes it to other algebraic groups, and connects it to knot homology and Hochschild homology of Soergel bimodules.

Contribution

It introduces a geometric framework for the Jones-Ocneanu trace applicable to all simple algebraic groups and proves related formality and freeness results.

Findings

The trace can be expanded in terms of irreducible characters.

Certain perverse sheaves are equivariantly formal.

Hochschild homology of Soergel bimodules is free.

Abstract

We give a geometric interpretation of the Jones-Ocneanu trace on the Hecke algebra, using the equivariant cohomology of sheaves on SL(n). This construction makes sense for all simple algebraic groups, so we obtain a generalization of the Jones-Ocneanu trace to Hecke algebras of other types. We give a geometric expansion of this trace in terms of the irreducible characters of the Hecke algebra, and conclude that it agrees with a trace defined independently by Gomi. Based on our proof, we also prove that certain simple perverse sheaves on a reductive algebraic group G are equivariantly formal for the conjugation action of a Borel B, or equivalently, that the Hochschild homology of any Soergel bimodule is free, as the authors had previously conjectured. This construction is closely tied to knot homology. This interpretation of the Jones-Ocneanu trace is a more elementary manifestation of the geometric construction of HOMFLYPT homology given by the authors in a previous paper.