  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.
