Positive Casimir and Central Characters of Split Real Quantum Groups

The notion of the positive representations was introduced in a joint work with Igor Frenkel as a new research program devoted to the representation theory of split real quantum groups. Explicit construction of the these irreducible representations have been made corresponding to classical Lie type. In this talk, I will discuss the action of the generalized Casimir operators, which is important to understand the tensor product decomposition of these representations. These operators are shown to admit positive eigenvalues, and that their image defines a semi-algebraic region bounded by real points of the discriminant variety.