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We describe the generalized Casimir operators and their actions on the positive representations  of the modular double of split real quantum groups . We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters . We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of , and we discuss explicit examples in the lower rank cases.

Positive Casimir and Central Characters of Split Real Quantum Groups