Positive Casimir and Central Characters of Split Real Quantum Groups

Abstract

We describe the generalized Casimir operators and their actions on the positive representations \({\mathcal{P}_\lambda}\) of the modular double of split real quantum groups \({\mathcal{U}_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})}\). We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters \({\lambda}\). We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of q, and we discuss explicit examples in the lower rank cases.