Carine Molitor-Braun

Représentations irréductibles bornées des groupes de Lie exponentiels

Let G be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations (T,U) of G on a Banach space U 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. Recu par les editeurs 12 novembre, 1999. Etude effectuee dans le cadre du projet de recherche MEN/CUL/98/007. Classification (AMS) par sujet: 43A20. Mots cles: groupe de Lie resoluble exponentiel, representation bornee topologiquement irreductible, orbite, norme d’extension, sous-espace invariant, ideal premier, ideal primitif. c ©Societe Mathematique du Canada 2001. 944

EXPONENTIAL ACTIONS, ORBITS AND THEIR KERNELS

(<&)).s i Somn 5(<9e) and Lpartial results are known for solvable exponential groups [13], but a lot of questionsremain open. In this paper we are therefore going to introduce an intermediate stepbetween nilpotent and exponential Lie groups: the exponential actions on nilpotentLie groups. Some special examples of exponential actions have already appeared inliterature. In [20] Poguntke considers a connected, simply connected exponential Liegroup <S = expg which acts in a natural way on 9t = expn, where n is the nilradicalof g. This induces an actio

Fine Disintegration of the Left Regular Representation

Let Hn be the (2n + 1)-dimensional Heisenberg group. We decompose L2(Hn) as the closure of a direct sum of infinitely many left translation invariant eigenspaces (for certain systems of partial differential equations). The restriction of the left regular representation to each one of these eigenspaces disintegrates into a direct integral of unitary irreducible representations, such that each infinite dimensional unitary irreducible representation appears with multiplicity 0 or 1 in this disintegration.

Spectral synthesis for flat orbits in the dual space of weighted group algebras of nilpotent Lie groups

Let G = exp(g) be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. We determine the structure of all the two-sided closed ideals of the weighted group algebra Lω(G) which are attached to a flat co-adjoint orbit.