Positive representations of non-simply-laced split real quantum groups

We construct the positive principal series representations for Uq(gR) where g is of type Bn, Cn, F4 or G2, parametrized by Rn where n is the rank of g. We show that under the representations, the generators of the Langlands dual group Uq˜(gRL) are related to the generators of Uq(gR) by the transcendental relations. This gives a new and very simple analytic relation between the Langlands dual pair. We define the modified quantum group Uqq˜(gR)=Uq(gR)⊗Uq˜(gRL) of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the q-tori polynomials.