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