Induced representations of the one-dimensional quantum Galilei group

We apply the induced representation technique to study the one dimensional quantum Galilei group. After a brief sketch of the general theory, we develop the representation at an algebraic level. Then we prove the existence of a quasi in invariant measure on the homogeneous space and the corresponding square integrable functions. The unitarity of the induced representations is first studied in the physically meaningful case of real quantum parameter, where the involution is not standard. The imaginary case, where $(\ast\circ S)^2=id$ exhibits a behaviour which is analogue to the classical one.