Ensemble des double-classes pour la désintégration des représenations des groupes de Lie nilpotents et noyaux d'opérateurs sur les groupes de lie exponentiels

Abstract : This thesis contains two independent parts. After recalling in chater 1 certain notions useful in the following, we begin the first part by the study of the set H \G /B of the double classes. Here, G = exp g is a nilpotent, connected and simply connected Lie group and H and B are two closed subgroups of G. Then, we devote chapter 2 to the description of the double classes which conduces us to concretize a " good " measure on this set. This is what we need in chapter 3 to desintegrate the restriction to the subgroup H of a unitary irreductible representation p de G which is induced from a character C 1 of B. We obtain explicitly the disintegration of pIh into irreducibles, and an intertwining isometric operator. As an application of this result, we study the tensor product of representations and we prove a necessary and sufficient condition for its irreducibility. The second part of the thesis is also about the representation theory of Lie groups. In fact, we study in chapter 4 the operator kernels for irreducible unitary representations of exponential Lie groups. A result appeared in a paper by Leptin will be discussed. We give counterexemple of a function F î C8 c (G/BxG/ ; C1) which can not be a kernel operator of any ? îL1(G), even though the polarization b = Lie B satisfies Leptin's hypothesis (since the given b is a tame polarisation). We conclude the last chapter considering the case where b is an ideal of g . Here, constructing a retract, we prove that Leptin's result becoms correct.
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Jawhar Abdennadher. Ensemble des double-classes pour la désintégration des représenations des groupes de Lie nilpotents et noyaux d'opérateurs sur les groupes de lie exponentiels. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 2004. Français. ⟨NNT : 2004METZ008S⟩. ⟨tel-01750237⟩

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