Geometry and Topology 12 (2008) 1243–1263

DOI: 10.2140/gt.2008.12.1243

Abstract

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.

Keywords

Soergel bimodule, Khovanov–Rozansky homology, Hochschild homology

Mathematical Subject Classification

Primary: 17B10

Secondary: 57T10

References

Publication

Received: 8 August 2007
Revised: 19 December 2007
Accepted: 15 March 2008
Published: 1 June 2008
Proposed: Vaughan Jones
Seconded: Rob Kirby, Ralph Cohen

Authors