Abstract

Let g = expg be a connected, simply connected, solvable exponential Lie group. Let l in g^* and let p be an appropriate Pukanszky polarization for l in g. For every p = (p₁,…,p_m) in [1,∞]^m we define a representation π_l,p,p by induction on an L^p-space, where the norm ∥•∥_p of this space is in fact obtained by successive L^p_j-norms, with distinct p_j’s in different directions. These representations are topologically irreducible and their restrictions to the subspaces generated by the vectors of the form π_l,p,p(f)ξ with f in L¹(g), π_l,p,p(f) of finite rank and ξ in H_l,p,p are algebraically irreducible. All the simple L¹(g)-modules are of that form, up to equivalence. We show that these representations may in fact be characterized (up to equivalence) by the g-orbits of couples (l,ν), where l in g^* and ν is a real linear form on g(l) ∕ g(l) ∩ n satisfying a certain growth condition and where g(l) is the stabilizer of l in g.

Authors