Characterization of the simple L¹(G)-modules for exponential Lie groups

Jean Ludwig, Salma Mint Elhacen and Carine Molitor-Braun

Vol. 212 (2003), No. 1, 133–156

DOI: 10.2140/pjm.2003.212.133

Abstract

Let 𝒢 = expg be a connected, simply connected, solvable exponential Lie group. Let l ∈ g^∗ and let p be an appropriate Pukanszky polarization for l in g. For every p = (p₁,…,p_m) ∈ [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 ∈ L¹(𝒢), π_l,p,p(f) of finite rank and ξ ∈ 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 ∈ 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.

Milestones

Received: 1 November 2000

Revised: 8 October 2002

Published: 1 November 2003