## Geometry & Topology

### A geometric model for Hochschild homology of Soergel bimodules

#### 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.

#### Article information

Source
Geom. Topol., Volume 12, Number 2 (2008), 1243-1263.

Dates
Revised: 19 December 2007
Accepted: 15 March 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800072

Digital Object Identifier
doi:10.2140/gt.2008.12.1243

Mathematical Reviews number (MathSciNet)
MR2425548

Zentralblatt MATH identifier
1198.20037

Subjects
Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 57T10: Homology and cohomology of Lie groups

#### Citation

Webster, Ben; Williamson, Geordie. A geometric model for Hochschild homology of Soergel bimodules. Geom. Topol. 12 (2008), no. 2, 1243--1263. doi:10.2140/gt.2008.12.1243. https://projecteuclid.org/euclid.gt/1513800072

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