Positive Casimir and Central Characters of Split Real Quantum Groups

Abstract

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.

33 pages, 6 figures Expanded introduction. Minor typo fixed

References in corpus (1)

• The quantum dilogarithm and representations quantum cluster varieties