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 (∞, 1)-category SBimE(n) of E-valued U(n) Soergel bimodules, where E is a connective E_∞-ring spectrum, and assemble them into a monoidal locally additive (∞, 2)-category SBimE. When E has a complex orientation, we then construct a braiding, i.e. an E₂-algebra structure, on the universal locally stable (∞, 2)-category Kᵇloc(SBimE) associated to SBimE. 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.