Unique characterization of the Fourier transform in the framework of representation theory

In this paper we elaborate upon the investigation initiated in [3] of typical and distinctive properties of the Fourier transform (FT), in particular the crucial role played by the Howe dual pair (O(m), sl2). We prove in detail a result on the unique characterization of the FT making extensive use of a representation of the Lie algebra sl2. As an example, we consider the case m = 1. We refer to [3] for a detailed study involving the derivation of a class of operators portraying FT symmetry properties.