Représentations tempérées des groupes de Lie nilpotents. (Tempered representations of nilpotent Lie groups) (Q913961)

Let G be a connected, simply connected nilpotent Lie group with Lie algebra \({\mathfrak g}\). It is well known that the unitary dual of G is fruitfully parameterized by Kirillov's orbit method. But our knowledge for non unitary representations is far from enough. The author develops a differentiable theory, instead of the unitary theory, based on his previous works [Invent. Math. 88, 375-394 (1987; Zbl 0595.43008)]. Let E be a G-module, namely a Fréchet space with continuous G-action. A vector \(x\in E\) is differentiable if the mapping \(G\ni g\to g\cdot x\in E\) is differentiable. We say E is differentiable if every \(x\in E\) is differentiable, and E is tempered if, for every continuous semi-norm p on E, there exist a continuous semi-norm q and \(m\in N\) such that \[ p(\exp (X)\cdot x)\leq (1+| X|)^ m q(x)\quad (\forall X\in {\mathfrak g},\quad \forall x\in E), \] where \(| \cdot |\) denotes a (arbitrary) norm on \({\mathfrak g}\). Besides many results on tempered representations, the author gives the classification of tempered differentiable G-modules which are topologically irreducible. More precisely, one of his main results is the following. Let E be a tempered differentiable G-module which is topologically irreducible. Then there is a unique irreducible unitary representation (\({\mathcal H},\pi)\) of G such that the space \({\mathcal H}^{\infty}\) of its \(C^{\infty}\)-vectors is isomorphic to E.