We give a geometric interpretation of the Jones-Ocneanu trace on the Hecke algebra using the equivariant cohomology of sheaves on . This construction makes sense for all simple 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 are equivariantly formal for the conjugation action of a Borel , or equivalently, that the Hochschild homology of any Soergel bimodule is free, as the authors had previously conjectured.
This construction is also 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.
Ben Webster. Geordie Williamson. "The geometry of Markov traces." Duke Math. J. 160 (2) 401 - 419, 1 November 2011. https://doi.org/10.1215/00127094-1444268