A geometric model for Hochschild homology of Soergel bimodules

Webster, Ben; Williamson, Geordie

doi:10.2140/gt.2008.12.1243

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A geometric model for Hochschild homology of Soergel bimodules

Ben Webster and Geordie Williamson

Geometry & 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 $B \times B$ –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 2000

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

Ben Webster
Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139
http://math.mit.edu/~bwebster
Geordie Williamson
Mathematisches Institut der Universität Freiburg Freiburg 79106 Germany
http://home.mathematik.uni-freiburg.de/geordie/

Volume 12, issue 2 (2008)