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A geometric model for Hochschild homology of Soergel bimodules

Ben Webster and Geordie Williamson

Geometry & Topology 12 (2008) 1243–1263

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.

Soergel bimodule, Khovanov–Rozansky homology, Hochschild homology
Mathematical Subject Classification 2000
Primary: 17B10
Secondary: 57T10
Received: 8 August 2007
Revised: 19 December 2007
Accepted: 15 March 2008
Published: 1 June 2008
Proposed: Vaughan Jones
Seconded: Rob Kirby, Ralph Cohen
Ben Webster
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
Geordie Williamson
Mathematisches Institut der Universität Freiburg
Freiburg 79106