Positive Representations of Split Real Simply-laced Quantum Groups

TL;DR

This paper constructs positive principal series representations for split real simply-laced quantum groups, explicitly describing generator actions, establishing transcendental relations, and exploring the structure of the modular double and its commutant.

Contribution

It introduces explicit positive representations for split real simply-laced quantum groups and analyzes their algebraic and analytical properties, including the modular double structure.

Findings

Explicit positive representations parametrized by non-negative reals

Transcendental relations between modular double generators

Embedding into quantum torus algebra and Langlands duality

Abstract

We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group $\mathbf{U}_{\mathfrak{q}\tilde{\mathfrak{q}}}(\mathfrak{g}_\mathbb{R})$ of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.