Positive Representations of Split Real Simply-Laced Quantum Groups

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.