Speaker

Ivan Ip (`ȋZw)

Date

October 8, 15:45-16:35

Title

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

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 \(A_n\).