Representations of the Affine Transformation Groups Acting Simply Transitively on Siegel Domains

Let G be the split solvable Lie group acting simply transitively on a Siegel domain D. We consider irreducible unitary representations of G realized on Hilbert spaces of holomorphic functions on D. We determine all such Hilbert spaces by connecting them with positive Riesz distributions on the dual cone and describe them through the Fourier–Laplace transform. Moreover we classify the representations of G by making use of the orbit method.