Induced Representations of the One-Dimensional Quantum Galilei Group

We apply the techniques of induced representations to study the one-dimensional quantum Galilei group. We prove the existence of a quasi-invariant measure on the homogeneous space and of the corresponding square integrable functions. First, the unitary property of the induced representations is studied in the physically meaningful case of a real quantum parameter where the involution is nonstandard. The behavior in the complex case, where (* S)²=id, is similar to the classical one. Bibliography: 5 titles.