Braiding on complex oriented Soergel bimodules

Abstract

In this note, we study U(n) Soergel bimodules in the context of stable homotopy theory. We define the $(\infty ,1)$-category ${\mathrm{SBim}}_E(n)$ of $E$-valued U(n) Soergel bimodules, where $E$ is a connective ${\mathbb{E}}_{\infty }$-ring spectrum, and assemble them into a monoidal locally additive $(\infty ,2)$-category ${\mathrm{SBim}}_E$. When $E$ has a complex orientation, we then construct a braiding, i.e. an ${\mathbb{E}}_2$-algebra structure, on the universal locally stable $(\infty ,2)$-category ${\mathrm{K}}_{\mathrm{loc}}^b({\mathrm{SBim}}_E)$ associated to ${\mathrm{SBim}}_E$. Along the way, we also prove spectral analogs of standard splittings of Soergel bimodules. This is a topological generalization of the type $A$ Soergel bimodule theory developed in a previous paper.

31 pages, comments welcome