Generalized Fourier Transform and Minimal representations (of p-adic groups)

The classical Fourier transform is an ubiquitous operator acting on L^2(V) for a finite-dimensional quadratic space V. We study it from the point of view of representation theory. Together with other operators it forms a remarkable representation of a metaplectic group on L^2(V), that has minimal functional dimension. Minimal representation of other groups, often have models on L^2(X) for a cone X. We shall see how to define generalized Fourier transforms on L^2(X) and discuss their properties.