#### Volume 12, issue 2 (2008)

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A geometric model for Hochschild homology of Soergel bimodules

### Ben Webster and Geordie Williamson

Geometry & Topology 12 (2008) 1243–1263
##### 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 $SL\left(n\right)$. We present a geometric model for this Hochschild homology for any simple group $G$, as $B$–equivariant intersection cohomology of $B\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}B$–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
Primary: 17B10
Secondary: 57T10