  We describe the generalized Casimir operators and their actions on the
positive representations $P_{\lambda}$ of the modular double of split real
quantum groups $U_{q\tilde{q}}(g_R)$. 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.
