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Espaces duaux de certains produits semi-directs et noyaux associés aux orbites plates

Abstract : The first problem treated in this thesis is the description of the dual topology 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. Our object is to compare the Fell topology of the dual space with the natural topology of the quotient space of admissible coadjoint orbits. The first example treated in this work is the case of the motion groups Mn = SO(n) x Rn. We have shown that the dual pace of Mn is homeomorphic with its admissible coadjoint orbit space. This result may be true also for the groups Gn = U(n) x Hn, where Hn is the 2n+1 dimensional Heisenberg Lie group (it is only proved for the group G1). The second issue regarded in this thesis is the determinaton of the irreducible unitary representation p of a group G, for which the kernel of p in L1(G) is given by the functions whose the Fourrier transform annihilates on the orbit O of p. This problem was solved for the case of nilpotent roups by J. Ludwig who had shown that ker(p) = {? ? L1(G); ?[accent circonflexe](Op) = {O}} if and only if Op is a flat orbit. The work is to prove that this result remains true for completely solvable Lie groups
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Submitted on : Thursday, March 29, 2018 - 1:52:47 PM
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Mounir Elloumi. Espaces duaux de certains produits semi-directs et noyaux associés aux orbites plates. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 2009. Français. ⟨NNT : 2009METZ009S⟩. ⟨tel-01752623⟩



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