Duke Mathematical Journal
- Duke Math. J.
- Volume 160, Number 2 (2011), 401-419.
The geometry of Markov traces
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.
Duke Math. J., Volume 160, Number 2 (2011), 401-419.
First available in Project Euclid: 27 October 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20C08: Hecke algebras and their representations
Secondary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials [See also 13Nxx, 32C38] 20G40: Linear algebraic groups over finite fields 16T99: None of the above, but in this section
Webster, Ben; Williamson, Geordie. The geometry of Markov traces. Duke Math. J. 160 (2011), no. 2, 401--419. doi:10.1215/00127094-1444268. https://projecteuclid.org/euclid.dmj/1319721315