A geometric model for Hochschild homology of Soergel bimodules

TL;DR

This paper introduces a geometric model for Hochschild homology of Soergel bimodules, linking it to intersection homology of orbit closures, and confirms a conjecture relating to its algebraic structure.

Contribution

It provides a geometric framework for Hochschild homology of Soergel bimodules across all simple groups, extending previous results and confirming a conjecture for type A.

Findings

Hochschild homology is modeled as equivariant intersection homology.

Orbit closures are equivariantly formal in type A.

Hochschild homology is an exterior algebra in smooth cases, with explicit degree and Hilbert series.

Abstract

An important step in the calculation of the triply graded link homology theory 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 equivariant intersection homology of B x B-orbit closures in G. We show that, in type A these orbit closures are equivariantly formal for the conjugation T-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 describe its Hilbert series, proving a conjecture of Jacob Rasmussen.