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 $$\mathcal {U}_q(\mathfrak {sl}(2,\mathbb {R}))$$ 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 $$\mathcal {U}_q(\mathfrak {g})$$ 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 $$A_{n-1}$$ A n - 1 degenerate representations of $$\mathcal {U}_q(\mathfrak {g}_\mathbb {R})$$ U q ( g R ) for general Lie types based on the complexification of the central parameters.