Fourier transforms on the basic affine space of a quasi-split group

Let G be an even orthogonal quasi-split group defined over a local non-archimedean field F. In the first part of the paper we describe the subspace of smooth vectors of the minimal representation of G(F), realized on the space of square-integrable functions on a cone. In the second part we use this description for an extension of the Gelfand and Graev construction of generalized Fourier transforms on basic affine space from split groups to quasi-split groups.