Abstract: 
In this talk, I will introduce the family of positive principal series representations for split real quantum groups by positive selfadjoint operators. The construction of these representations gives the starting point of a new research program devoted to the representation theory of split real quantum groups initiated in the joint work with Igor Frenkel. It is a generalization of the special class of representations considered by J. Teschner for Uq(sl(2,R)) in Liouville theory, where it exhibits a strong parallel to the finitedimensional representation theory of quantum groups. Recently from the construction of the positive representations, a direct analytic relation between modular duality and Langlands duality is also discovered, which should have deep consequences in the Langlands program.
