Duke Mathematical Journal

The geometry of Markov traces

Ben Webster and Geordie Williamson

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Abstract

We give a geometric interpretation of the Jones-Ocneanu trace on the Hecke algebra using the equivariant cohomology of sheaves on SLn. 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 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 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.

Article information

Source
Duke Math. J., Volume 160, Number 2 (2011), 401-419.

Dates
First available in Project Euclid: 27 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1319721315

Digital Object Identifier
doi:10.1215/00127094-1444268

Mathematical Reviews number (MathSciNet)
MR2852120

Zentralblatt MATH identifier
1254.20009

Subjects
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

Citation

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


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