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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
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