Let Open image in new window be an exponential solvable Lie group. In this chapter we characterize bounded, topologically irreducible Banach-space representations of G using triples ( Ω, τ, ∥∥), where Open image in new window is a coadjoint orbit of G, τ is a topologically irreducible representation of the algebra \(L^1({\mathbb R}^n,\omega ) \) for a certain \(n\in {\mathbb N}^* \) and a weight ω on \({\mathbb R}^n \), and ∥∥ is a so-called extension norm.