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Nous donnons dans cet article une désintégration en irréductibles explicite des
restrictions aux sous-groupes connexes fermés des représentations unitaires et
irréductibles pour les groupes de Lie nilpotents simplement connexes. Ainsi, nous
décrivons un opérateur d'entrelacement qui ne tient pas compte des multiplicités
intervenant dans la désintégration.
Let G be an exponential solvable Lie group, H and A two closed connected subgroups of G and σ a unitary and irreducible representation of H. We prove the orbital spectrum formula of the Up-Down representation ρ(G, H, A, σ) = Ind
σ. When G is nilpotent, the multiplicities of such representation turns out to be uniformly infinite or finite and bounded. A necessary and sufficient condition for the finiteness of the multiplicities is given. The same results are obtained when G is exponential...
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