Verma Howe duality and LKB representations

TL;DR

This paper introduces a novel version of Howe duality involving Verma modules, extends it through quantization, and applies it to demonstrate the simplicity of LKB representations for braid groups.

Contribution

It establishes a new duality framework with Verma modules, quantizes it, and connects it to the structure of LKB representations and braid group modules.

Findings

LKB representations are simple modules for braid groups.

A new duality involving tensor products of Verma modules is established.

Quantization of this duality is achieved.

Abstract

We establish a version of Howe duality that involves a tensor product of Verma modules. Surprisingly, this duality leaves the realm of lowest and highest weight modules. We quantize this duality, and as an application, we prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for the braid group and for various of its subgroups, including the pure braid group.