AbstractLet G be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations (T, ) of G on a Banach space by giving a G-orbit in n* (n being the nilradical of g), a topologically irreducible representation of L1(ℝn, ω), for a certain weight ω and a certain n ∈ ℕ, and a topologically simple extension norm. If G is not symmetric, i.e., if the weight ω is exponential, we get a new type of representations which are fundamentally different from the induced representations.
Representations of a semisimple Lie group on a Banach space. I
