Positive Representations of Split Real Simply-laced Quantum Groups

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 ${\mathbb{R}}_{\ge 0}^r$ 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.