Représentation étoile du revêtement universel du groupe hyperbolique et formule de Plancherel

Abstract : In this thesis, we apply a program of description and construction of irreductible unitary representation (I.U.R) of Lies groups by star deformation, to the universal covering group of the hyperbolic group. Let us denote by G this group. In chapter 1, we recall the definitions associated to the group G, and its I.U.R which are traditionally classified in three series : principal series, discrete series and complementary series. In chapter 2, we describe the representations of principales series using a star product equivalent to the Moyal product. We then define a star exponential and the restriction of the adapted Fourier transformation to an orbit. In chapter 3, we define and describe the deformation of the algebra of functions on the orbits associated to the representations of the discrete series of G, and we recall the construction of coherent states and Berezin symboles and corresponding star product. In chapter 4, we realize the complementary series of G as a star representation using a Moyal product on RxT and realize of the Lie algebra g of G as complex functions on RxT, wich is deformation of parametrisation of the cone. In chapter 5, we choose a Lebesgue measure on the dual g* of G and define a global adapted Fourier transformation E by glueing to guether the two constructions associated to principal series and discrete series. Finally, using the Plancherel formula for G, we show that the adapted Fourier transformation is a unitary transformation bitween L2 (G) and a space of square star integrable on g*, and that the plancharel formula is obtained by inverting E.
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Mohamed Yahyai. Représentation étoile du revêtement universel du groupe hyperbolique et formule de Plancherel. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1995. Français. ⟨NNT : 1995METZ001S⟩. ⟨tel-01776868⟩

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