  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.
