C*-modules et opérateurs d'entrelacement associés à la série principale de groupes de Lie semi-simples

Pierre Clare 1
1 Laboratoire mathématiques - analyse, probabilités, modélisation
MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
Abstract : This thesis is devoted to the study of the unitary principal series of certain semisimple Lie groups, within the framework of non-commutative geometry. For a family of minimal parabolic subgroups sharing the same Levi component L, we describe the associated unitary principal series representations by means of C*(L)-Hilbert modules. This construction is inspired from the work of M. A. Rieffel and we provide different realisations for the modules that it yields, thus translating at a global level the classical pictures of the principal series. For real-rank 1 groups, we characterise a certain class of bounded operators on those modules, and obtain an irreducibility result, analogous to Bruhat's classical one. We then establish the convergence, on certain submodules, of intertwining integrals close to the ones defining Knapp and Stein operators. Those integrals can be written as the sum of a densely defined and likely bounded operator, a densely defined unbounded operator and a residual term. We finally indicate, in special cases, a normalisation process which yields unitary intertwining operators between Hilbert modules. Those operators implement the Weyl group action related to unitary equivalences among the principal series at the level of the group reduced C*-algebra.
Document type :
Mathématiques [math]. Université d'Orléans, 2009. Français. <NNT : 2009ORLE2028>

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Pierre Clare. C*-modules et opérateurs d'entrelacement associés à la série principale de groupes de Lie semi-simples. Mathématiques [math]. Université d'Orléans, 2009. Français. <NNT : 2009ORLE2028>. <tel-00454669>




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