A normalized family of representations of the group of motions of a Euclidean space and the inverse problem of the representation theory of this group

There exists a well-known holomorphic family Tλ of representations of the isometry group of R such that T-λ∼Tλ for λ≠0. This paper presents a holomorphic family VRλ, |λ|<R, such that VRλ∼Tλ and VR-λ= VRλ for λ≠0. It is used for the construction of (generally speaking, reducible) representations of a fairly general form.