A geometric model for Hochschild homology of Soergel bimodules

Webster, Ben¹ ; Williamson, Geordie²

¹ Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139
² Mathematisches Institut der Universität Freiburg, Freiburg 79106, Germany

Geometry & topology, Tome 12 (2008) no. 2, pp. 1243-1263.

Résumé

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 $SL\left(n\right)$. 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.

Détail

DOI : 10.2140/gt.2008.12.1243

Keywords: Soergel bimodule, Khovanov–Rozansky homology, Hochschild homology

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