×

zbMATH — the first resource for mathematics

Induced representations of completely solvable Lie groups. (English) Zbl 0724.22006
On the line of his research started by [Trans. Am. Math. Soc. 313, 433- 473 (1989; Zbl 0683.22009)], the author gives here the explicit decomposition into irreducible constituents for induced representations of completely solvable Lie groups. The results are described in terms of the orbit method and generalize those obtained in the nilpotent case by L. Corwin, F. Greenleaf and G. Grélaud [ibid. 304, 549-583 (1987; Zbl 0629.22005)]. Let G be a connected and simply connected completely solvable Lie group with Lie algebra \({\mathfrak g}\), and let H be a connected subgroup of G with Lie algebra \({\mathfrak h}\). Given \(\phi\in {\mathfrak g}^*\), we construct an irreducible unitary representation \(\pi_{\phi}\) of G. This construction leads us to the fact that the unitary dual \(\hat G\) of G can be parametrized by the space of coadjoint orbits of G [cf. P. Bernat et al., Représentations des groupes de Lie résolubles, Dunod, Paris (1972; Zbl 0248.22012)]. For \(\pi\in \hat G\), we denote by \(O_{\pi}\) the corresponding coadjoint G-orbit. Take \(\nu\in \hat H\). Let p: \({\mathfrak g}^*\to {\mathfrak h}^*\) be the canonical projection. We consider the natural measure on \(p^{- 1}(O_{\nu})\), which is the fiber measure with H-invariant measure on the base \(O_{\nu}\) and Lebesgue measure on the affine fiber \({\mathfrak h}^{\perp}=\{\phi \in {\mathfrak g}^*\); \(\phi |_{{\mathfrak h}}=0\}\). Then we have the following decomposition formula: \[ ind^ G_ H \nu =\int^{\oplus}_{G\cdot p^{- 1}(O_{\nu})/G}n_{\phi}\pi_{\phi}d\mu ({\dot \phi}), \] where \(\mu\) is the push-forward of the natural measure under \(p^{-1}(O_{\nu})\to G\cdot p^{-1}(O_{\nu})/G\) and the multiplicity \(n_{\phi}\) is given by the number of H-orbits contained in \(G\cdot \phi \cap p^{- 1}(O_{\nu})\).

MSC:
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22E25 Nilpotent and solvable Lie groups
22D30 Induced representations for locally compact groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] P. Bernat , Sur les représentations unitaires des groupes de Lie résolubles , Ann. Sci. Ecole Norm. Sup. , 82 ( 1965 ), 37 - 99 . Numdam | MR 194553 | Zbl 0138.07302 · Zbl 0138.07302 · numdam:ASENS_1965_3_82_1_37_0 · eudml:81805
[2] P. Bernat , et al., Représentations des groupes de Lie résolubles , Dunod , Paris , 1972 . MR 444836 | Zbl 0248.22012 · Zbl 0248.22012
[3] I. Butsyatskaya , Representations of exponential Lie groups , Functional Anal. Appl. , 7 ( 1973 ), 151 - 152 . MR 325855 | Zbl 0286.22013 · Zbl 0286.22013 · doi:10.1007/BF01078888
[4] L. Corwin - F. Greenleaf - G. Grelaud , Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups , Trans. Amer. Math. Soc. , 304 ( 1987 ), 549 - 583 . MR 911085 | Zbl 0629.22005 · Zbl 0629.22005 · doi:10.2307/2000731
[5] L. Corwin - F. Greenleaf , Complex algebraic geometry and calculation of multiplicities for induced representations of nilpotent Lie groups , Trans. Amer. Math. Soc. , 305 ( 1988 ), 601 - 622 . MR 924771 | Zbl 0651.22004 · Zbl 0651.22004 · doi:10.2307/2000880
[6] A. Kirillov , Unitary representations of nilpotent Lie groups , Russian Math. Surveys , 17 ( 1962 ), 53 - 104 . MR 142001 | Zbl 0106.25001 · Zbl 0106.25001 · doi:10.1070/rm1962v017n04ABEH004118
[7] A. Kleppner - R. Lipsman , The Plancherel formula for group extensions I , Ann. Sci. Ecole Norm. Sup. , 5 ( 1972 ), 459 - 516 . Numdam | MR 342641 | Zbl 0239.43003 · Zbl 0239.43003 · numdam:ASENS_1972_4_5_3_459_0 · eudml:81904
[8] R. Lipsman , Characters of Lie groups II, J . Analyse Math. , 31 ( 1977 ), 257 - 286 . MR 579006 | Zbl 0351.22009 · Zbl 0351.22009 · doi:10.1007/BF02813305
[9] R. Lipsman , Orbital parameters for induced and restricted representations , Trans. Amer. Math. Soc. , 313 ( 1989 ), 433 - 473 . MR 930083 | Zbl 0683.22009 · Zbl 0683.22009 · doi:10.2307/2001416
[10] R. Lipsman , Harmonic analysis on exponential solvable homogeneous spaces: The algebraic or symmetric cases, Pacific J . Math. , 140 ( 1989 ), 117 - 147 . Article | MR 1019070 | Zbl 0645.43010 · Zbl 0645.43010 · doi:10.2140/pjm.1989.140.117 · minidml.mathdoc.fr
[11] G. Mackey , Unitary representations of group extensions , Acta Math. , 99 ( 1958 ), 265 - 311 . MR 98328 | Zbl 0082.11301 · Zbl 0082.11301 · doi:10.1007/BF02392428
[12] L. Pukanszky , Unitary representations of Lie groups with co-compact radical and applications , Trans Amer. Math. Soc. , 236 ( 1978 ), 1 - 49 . MR 486313 | Zbl 0389.22009 · Zbl 0389.22009 · doi:10.2307/1997773
[13] O. Takenouchi , Sur la facteur représentation d’une groupe de Lie résoluble de type (E) , Math. J. Okayama Univ. , 7 ( 1957 ), 151 - 161 . MR 97464 | Zbl 0080.02302 · Zbl 0080.02302
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.