×

zbMATH — the first resource for mathematics

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:
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22E30 Analysis on real and complex Lie groups
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Désintégration des représentations monomiales des groupes de Lie nilpotents, Journal of Lie Theory, 9, 1, 157-191, (1999) · Zbl 0921.22006
[2] Dual topology of a nilpotent Lie group, Ann. Sci. École Normale Sup., 6, 407-411, (1973) · Zbl 0284.57026
[3] A canonical approach to multiplicity formulas for induced and restricted representations of nilpotent Lie groups, Communications on Pure and Applied Mathematics, XLI, 1051-1088, (1988) · Zbl 0667.22004
[4] Spectrum and multiplicities for restrictions of unitary representation in nilpotent Lie groups, Pacific Journal of Mathematics, 135, 2, 233-267, (1988) · Zbl 0628.22007
[5] Sur LES restrictions des représentations unitaires des groupes de Lie résolubles exponentiels, Invent. Math., 104, 647-654, (1991) · Zbl 0702.22012
[6] On the connection between nilpotent groups and oscillatory integrals associated to singularities, Pacific Journal of Mathematics, 73, 329-364, (1977) · Zbl 0383.22009
[7] On the nilpotent \(⋆ \)- Fourier transform, Letters in Mathematical Physics, 30, 23-34, (1994) · Zbl 0798.22004
[8] On representations of simply connected nilpotent and solvable Lie groups, (1993)
[9] Unitary representations of nilpotent Lie groups, Uspekhi Mat. Nauk., 17, 4, 57-110, (1962) · Zbl 0106.25001
[10] Intégrale d’entrelacement sur des groupes de Lie nilpotents et indices de Maslov, Lect. Notes in Math., 587, 160-176, (1977) · Zbl 0391.22008
[11] Orbital parameters for induced and restricted representation, Trans. Amer. Math. Soc., 313, 433-473, (1989) · Zbl 0683.22009
[12] Restricting representations of completely solvable Lie groups, Can. J. Math., XLII, 5, 790-824, (1990) · Zbl 0724.22007
[13] Leçons sur les représentations des groupes, (1967), Dunod, Paris · Zbl 0152.01201
[14] Construction de sous-algèbres subordonnées à un élément du dual d’une algèbre de Lie résoluble, C. R. Acad. Sci., (Paris), 270, 173-175, (1970) · Zbl 0209.06701
[15] Construction de sous-algèbres subordonnées à un élément du dual d’une algèbre de Lie résoluble, C. R. Acad. Sci., (Paris), 270, 704-707 · Zbl 0209.06702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.