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

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 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 $A_n$.

arXiv comment62 pages, 12 figures

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