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

Parabolic Positive Representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$