Positive Representations with Zero Casimirs

TL;DR

This paper introduces a new family of positive representations for split-real quantum groups where Casimir operators degenerate to zero, revealing novel tensor product structures and explicit Casimir actions across Lie types.

Contribution

It constructs a new class of positive representations with zero Casimirs, extending the understanding of quantum group representations and their tensor decompositions.

Findings

New family of positive representations with zero Casimirs

Surprising tensor product decomposition observed

Explicit Casimir actions computed for various Lie types

Abstract

In this paper, we construct a new family of generalization of the positive representations of split-real quantum groups based on the degeneration of the Casimir operators acting as zero on some Hilbert spaces. It is motivated by a new observation arising from modifying the representation in the simplest case of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ compatible with Faddeev's modular double, while having a surprising tensor product decomposition. For higher rank, the representations are obtained by the polarization of Chevalley generators of $\mathcal{U}_q(\mathfrak{g})$ in a new realization as universally Laurent polynomials of a certain skew-symmetrizable quantum cluster algebra. We also calculate explicitly the Casimir actions of the maximal $A_{n-1}$ degenerate representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ for general Lie types based on the complexification of the central parameters.