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Entrelacement des restrictions des représentations unitaires des groupes de Lie nilpotents. (Intertwining of restriction of unitary representations of nilpotent Lie groups). (French) Zbl 0972.22006
Let $$G=\exp {\mathfrak g}$$ be a connected, simply connected nilpotent Lie group with Lie algebra $${\mathfrak g},H= \exp{\mathfrak h}$$ its analytic subgroup with Lie algebra $${\mathfrak h}$$. We denote by $$\widehat G$$ the unitary dual of $$G$$ and consider an irreducible unitary representation $$\pi$$ of $$G$$, namely $$\pi\in\widehat G$$ under the identification of each irreducible unitary representation with its equivalence class. Now the restriction $$\pi|_H$$ of $$\pi$$ onto $$H$$ is decomposed into irreducibles as follows. Let $$\Omega(\pi)$$ be the coadjoint orbit of $$G$$ corresponding to $$\pi\in \widehat G$$ and $$\mu$$ be a finite measure on $$\Omega(\pi)$$ equivalent to a $$G$$-invariant measure. Composing two mappings, the projection $$p:{\mathfrak g}^* \to{\mathfrak h}^*$$ and the Kirillov mapping $$\theta: {\mathfrak h}^* \to\widehat H$$, we take the image $$\nu$$ of $$\mu$$, i.e. $$\nu=(\theta \circ p)_*(\mu)$$ on the unitary dual $$\widehat H$$ of $$H$$. For $$\sigma\in \widehat H$$, we denote by $$\omega(\sigma)$$ the corresponding coadjoint orbit of $$H$$ and by $$m(\sigma)$$ the number of $$H$$-orbits contained in $$\Omega (\pi) \cap p^{-1} (\omega(\sigma))$$. Then it is well known [L. Corwin and F. P. Greenleaf, Pac. J. Math. 135, 233-267 (1988; Zbl 0628.22007)] that $$\pi|_H \simeq\int^\oplus_{\widehat H}m(\sigma) \sigma d\nu(\sigma)$$.
This paper begins by making concrete this decomposition through the determination of a base space of decomposition. Then, using this result the authors give an explicit intertwining operator between the restriction $$\pi|_H$$ and its concrete decomposition into the irreducibles just obtained. The whole work is pursued in the framework of the orbit method [cf. A. A. Kirillov, Usp. Mat. Nauk 17, 57-110 (1962; Zbl 0106.25001); L. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their applications, Part I, Cambridge/New York (1990; Zbl 0704.22007)].

##### MSC:
 2.2e+28 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 2.2e+31 Analysis on real and complex Lie groups
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