Braiding on complex oriented Soergel bimodules

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 \(\mathrm{K}^b_{\mathrm{loc}}(\mathrm{SBim}_E)\) 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.