Geometry & Topology

A geometric model for Hochschild homology of Soergel bimodules

Ben Webster and Geordie Williamson

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An important step in the calculation of the triply graded link homology of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL(n). We present a geometric model for this Hochschild homology for any simple group G, as B–equivariant intersection cohomology of B×B–orbit closures in G. We show that, in type A, these orbit closures are equivariantly formal for the conjugation B–action. We use this fact to show that, in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and to describe its Hilbert series, proving a conjecture of Jacob Rasmussen.

Article information

Geom. Topol., Volume 12, Number 2 (2008), 1243-1263.

Received: 8 August 2007
Revised: 19 December 2007
Accepted: 15 March 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 57T10: Homology and cohomology of Lie groups

Soergel bimodule Khovanov–Rozansky homology Hochschild homology


Webster, Ben; Williamson, Geordie. A geometric model for Hochschild homology of Soergel bimodules. Geom. Topol. 12 (2008), no. 2, 1243--1263. doi:10.2140/gt.2008.12.1243.

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