Positive Representations with Zero Casimirs

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 Uq (sl (2, R) ) U q (sl (2, 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 Uq (g) U q (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₍-₁ A n - 1 degenerate representations of Uq (g_ R) U q (g R) for general Lie types based on the complexification of the central parameters.