  We construct a new family of irreducible representations of
$\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ and its modular double by quantizing
the classical parabolic induction corresponding to arbitrary parabolic
subgroups, such that the generators of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$
act by positive self-adjoint operators on a Hilbert space. This generalizes the
well-established positive representations which corresponds to induction by the
minimal parabolic (i.e. Borel) subgroup. We also study in detail the special
case of type $A_n$ acting on $L^2(\mathbb{R}^n)$ with minimal functional
dimension, and establish the properties of its central characters and universal
$\mathcal{R}$ operator. We construct a positive version of the evaluation
module of the affine quantum group
$\mathcal{U}_q(\widehat{\mathfrak{sl}}_{n+1})$ modeled over this minimal
positive representation of type $A_n$.
