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Fonction de multiplicité de Corwin-Greenleaf pour certains groupes de Lie à nilradical co-compact

Abstract : Let G ⊃ H be Lie groups, g ⊃ h their Lie algebras, and pr : g∗ → h∗ the natural projection. For coadjoint orbits O^G ⊂ g∗ and O^H ⊂ h∗, we denote by n(O^G, O^H) the number of H-orbits in the intersection O^G ∩pr^−1(O^H), which is known as the Corwin-Greenleaf multiplicity function Our goal in this thesis is the description of the Corwin-Greenleaf multiplicity function of Lie groups with co-compact nilpotent radical, in particular the semi direct products G = K x N of compacts groups K with nilpotent Lie groups N. The dual space ^G of G had been determined via Mackey’s theory and the geometric parametrization given by R. L. Lipsmann who had proved that there is a bijection between ^G and the admissible coadjoint orbit space of G. In the spirit of the orbit method due to Kirillov and Kostant, one expects that n(O^G, O^K) coincides with the multiplicity of τ ∈ ^K occurring in an irreducible unitary representation π of G when restricted to K, if π is attached to O^G and τ is attached to O^K. The first example treated in this work is the case of the compact extensions of the group IR^n, we investigate the relationship between n(O^G;O^K) and the multiplicity m(π; τ ) of τ in the restriction of π to K. If π is infinite dimensional and the associated little group is connected, we show that n(O^G;O^K) ≠ 0 if and only if m(π; τ ) ≠ 0. Furthermore, for K = SO(n), n > 2, we give a sufficient condition on the representations π and τ in order that n(O^G;O^K) = m(π; τ ). The second example regarded in this thesis is the case of the compact extensions of the Heisenberg group IHn, we give two sufficient conditions on O^G in order that n(O^G;O^K) =< 1 for any K-coadjoint orbit O^K ⊂ k*. For K = U(n), assuming furthermore that O^G and O^K are admissible and denoting respectively by π and τ their corresponding irreducible unitary representations, we also discuss the relationship between n(O^G;O^K) and the multiplicity m(π; τ ) of π in the restriction of τ to K. Especially, we study in Theorem 4 the case where n(O^G;O^K) ≠ m(π; τ ). This inequality is interesting because we expect the equality as the naming of the Corwin-Greenleaf multiplicity function suggests.
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Anis Messaoud. Fonction de multiplicité de Corwin-Greenleaf pour certains groupes de Lie à nilradical co-compact. Mathématiques [math]. Laboratoire de Mathématiques Appliquées et Analyse Harmonique, 2018. Français. ⟨tel-01839302v2⟩

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