Braiding on complex oriented Soergel bimodules

TL;DR

This paper develops a topological framework for U(n) Soergel bimodules within stable homotopy theory, introducing a braiding structure when the spectrum has a complex orientation, and generalizing classical algebraic results.

Contribution

It constructs a braiding (E2-structure) on the universal stable category of U(n) Soergel bimodules in a topological setting, extending type A theory to spectral contexts.

Findings

Defined the $( abla, 1)$-category of $E$-valued U(n) Soergel bimodules.

Constructed a braiding (E2-structure) when $E$ has a complex orientation.

Proved spectral analogs of standard bimodule splittings.

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