<p>This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries.</p><p>The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions.</p><p>The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.</p>

This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries. The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions. The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.

L'objet de cette thèse est l'étude de certaines classes de variétés complexes compactes non kählériennes. On regarde d'abord la classe des surfaces de Kato. Étant donnés une surface de Kato minimale S, D le diviseur maximal de S formé des courbes rationnelles de S et ϖ : Š ͢ S le revêtement universel de S, on démontre que Š \ϖ-1 (D) est une variété de Stein. Les variétés LVMB sont la seconde classe de variétés non kählériennes étudiées. Ces variétés complexes sont obtenues en quotientant un ouvert U de Pn par un sous-groupe de Lie fermé G de (C*)n de dimension m. On reformule ce procédé en remplaçant U par la donnée d'un sous-éventail de celui de Pn et G par un sous-espace vectoriel de Rn convenable. On construit ensuite de nouvelles variétés complexes compactes non kählériennes en combinant une méthode due à Sankaran et celle donnant les variétés LVMB. Sankaran considère un ouvert U d'une variété torique dont le quotient par un groupe W discret est une variété compacte. Ici, on munit une certaine variété torique Y de l'action d'un sous-groupe de Lie G de (C*)n de sorte que le quotient X de Y par G soit une variété, puis on quotiente un ouvert de X par un groupe discret W analogue à celui de Sankaran.Enfin, on étudie les variétés OT, une autre classe de variétés non kählériennes, dont on démontre que leur dimension algébrique est nulle. Ces variétés sont obtenues comme quotient d'un ouvert de Cm par le produit semi-direct du réseau des entiers d'une extension de corps finie K de Q et d'un sous-groupe des unités de K bien choisi. / In this thesis we study certain classes of complex compact non-Kähler manifolds. We first look at the class of Kato surfaces. Given a minimal Kato surface S, D the divisor consisting of all rational curves of S and ϖ : Š ͢ S the universal covering of S, we show that Š \ϖ-1 (D) is a Stein manifold. LVMB manifolds are the second class of non-Kähler manifolds that we study here. These complex compact manifolds are obtained as quotient of an open subset U of Pn by a closed Lie subgroup G of (C*)n of dimension m. We reformulate this procedure by replacing U by the choice of a subfan of the fan of Pn and G by a suitable vector subspace of R^{n}. We then build new complex compact non Kähler manifolds by combining a method of Sankaran and the one giving LVMB manifolds. Sankaran considers an open subset U of a toric manifold whose quotient by a discrete group W is a compact manifold. Here, we endow some toric manifold Y with the action of a Lie subgroup G of (C^{*})^{n} such that the quotient X of Y by G is a manifold, and we take the quotient of an open subset of X by a discrete group W similar to Sankaran's one.Finally, we consider OT manifolds, another class of non-Kähler manifolds, and we show that their algebraic dimension is 0. These manifolds are obtained as quotient of an open subset of C^{m} by the semi-direct product of the lattice of integers of a finite field extension K over Q and a subgroup of units of K well-chosen.