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.
