Président du jury Willem, Michel
Promoteur Bertelson, Mélanie
;Bieliavsky, Pierre Publication Non publié, 2015-04-23
| Résumé : | This thesis provides the explicit expression of the star-exponential for the action of normal j-groups on their coadjoint orbits, and of the so-called modified star-exponential defined by Gayral et al. Using this modified star-exponential as the kernel of a functional transform between the group and its coadjoint orbits yields an adapted Fourier transform which is also detailed here. The normal j-groups arise in the work of Pytatetskii-Shapiro, who established the one-to-one correspondence with homogeneous bounded domains of the complex space; these groups are also the central element of the deformation formula recently developed by Bieliavsky & Gayral (a non abelian analog of the strict deformation quantization theory of Rieffel). Since these groups are exponential, the results given in this text illustrate the general work of Arnal & Cortet on the star-representations of exponential groups. As this work is meant to be as self-contained as possible, the first chapter reproduces many definitions introduced by Bieliavsky & Gayral, in order to obtain the expression of the symplectic symmetric space structure on normal j-groups, and of their unitary irreducible representations. The Weyl-type quantizer associated to this symmetric structure is then computed, thus yielding the Weyl quantization map for which the composition of symbols is precisely the deformed product defined by Bieliavsky-Gayral on normal j-groups. A detailed proof of the structure theorem of normal j-groups is also provided. The second chapter focuses on the expression and properties of the star-exponential itself, and exhibits a useful tool for the computation, namely the resolution of the identity associated to square integrable unitary irreducible representations of the groups. The result thus obtained satisfies the usual integro-differential equation defining the star-exponential. A criterion for the existence of a tempered pair underlying a given tempered structure on Lie groups is proven; the star-exponential functions are also shown to belong to the multiplier algebra of the Schwartz space associated to the tempered structure. Before that, it is shown that all Schwartz spaces that appear in this work are isomorphic as topological vector spaces. The modified version of this star-exponential is computed in chapter three, first for elementary normal j-groups and then for normal j-groups. It is then used to define an adapted Fourier transform between the group and the dual of its Lie algebra. This transform generalizes (to all normal j-groups) a Fourier transform that was already studied in the “ax+b” case by Gayral et al. (2008), as well as by Ali et al. (2003) in the context of wavelet transforms. |
