Abstract: 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 introduced by [Frenkel-Ip] which correspond 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. Finally we will explain 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.
Ref: arXiv:2008.08589