Geometry & Topology

A geometric model for Hochschild homology of Soergel bimodules

Ben Webster and Geordie Williamson

Full-text: Open access

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
Received: 8 August 2007
Revised: 19 December 2007
Accepted: 15 March 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Keywords
Soergel bimodule Khovanov–Rozansky homology Hochschild homology

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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