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.

Legendrian knots and constructible sheaves

“…A proof of this statement is given in [91, §6.3], using [90]. Here we give an expanded treatment, specialized to the case of the usual (uncolored) HOMFLY homology, for the purpose of spelling things out in enough detail to explicitly match gradings.…”

Diagrammatics for Soergel Categories

“…For more information on Soergel bimodules and their applications we refer the reader to [42,43,44,15,23,29,30,39,50,52] and references therein.…”

A diagrammatic categorification of the $q$-Schur algebra

“…Their work still remains to be extended to include the Hochschild homology. Besides this approach, which is the one most directly related to the results in this paper, we should also mention a geometric approach due to Webster and Williamson in [37] and a representation theoretic approach due to Mazorchuk and Stroppel [29].…”