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

TL;DR

This paper introduces a new family of positive, irreducible representations of quantum groups associated with real Lie algebras, generalizing previous models by incorporating arbitrary parabolic subgroups and analyzing their properties.

Contribution

It constructs a broad class of positive representations via parabolic induction, extending the known minimal parabolic case, and explores their structure, especially for type A, including their central characters and R-matrix.

Findings

Constructed irreducible positive representations for arbitrary parabolic subgroups.

Analyzed the type A case with minimal functional dimension.

Developed a positive evaluation module for affine quantum groups.

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 $A_n$.