This licentiate thesis consists of one paper about unitary representationtheory of ample groupoids and semigroups together with generalizationsto étale and non-Hausdorff groupoids. In the paper we study algebraically the type I and CCR properties forample Hausdorff groupoids. Clarke and Van Wyk proved that both ofthese properties admit a topological characterization for Hausdorff second countable groupoids in terms of separation properties of their orbitspace and the isotropy groups. Using a Stone type duality between ample groupoids and Boolean inverse semigroups with meets, we exploit thischaracterization to get a purely algebraic statement. We also apply thoseresults to get characterizations of the type I and CCR properties for inverse semigroups using their Boolean inverse completions. The generalization is about characterizing the same properties for both étale and ample non-necessarily Hausdorff groupoids which nonethelesshave Hausdorff unit spaces. In this setup, we first give a direct proofof the topological characterization for the CCR property which doesn't rely on the disintegration theory. The argument cannot be adapted toget an easier proof in the type I case, but we rather explain how to geta proof following the original ideas of Clark and Van Wyk in that case.Finally, we state for both étale and ample groupoids algebraic conditionsequivalent to the CCR and GCR properties on their pseudogroup of openand compact open bisections respectively.

Ce travail s’inscrit dans l’étude des symétries d’espaces de dimension infinie. Il répond à des questions algébriques en suivant des méthodes analytiques. Plus précisément, nous étudions certaines représentations du groupe symplectique complexe dans des espaces fonctionnels. Elles sont caractérisées par leurs décompositions isotypiques relativement à un sous-groupe compact maximal. Ce travail décrit ces décompositions dans deux modèles : un modèle classique (dit compact) et un autre plus récent (dit non-standard). Nous montrons que cela établit un lien entre deux familles de fonctions spéciales (fonctions hypergéométriques et fonctions de Bessel) ; ces familles sont associées à des équations différentielles ordinaires d’ordre 2, fuchsiennes dans un cas et non fuchsiennes dans l’autre. Nous mettons aussi en évidence, dans le modèle non-standard, un lien avec certaines équations d'Emden-Fowler, ainsi qu’un opérateur différentiel simple qui agit sur les décompositions isotypiques. / The general setting of this work is the study of symmetry groups of infinite-dimensional spaces. We answer algebraic questions, using analytical methods. To be more specific, we study certain representations of the complex symplectic group in functional spaces. These representations are characterised by their isotypic decompositions with respect to a maximal compact subgroup. In this work, we describe these decompositions in two different models: a classical model (compact picture) and a more recent one (non-standard picture). We show that this establishes a connection between two families of special functions (hypergeometric functions and Bessel functions); these families correspond to second order differential equations, which are Fuchsian in one case and non-Fuchsian in the other. We also establish a link with certain Emden-Fowler equations and exhibit a simple differential operator that acts on the isotypic decompositions.

In the thesis we acquaint ourselves with the terms from ergodic theory and re- presentation theory of topological groups. We pay attention particularly to terms unitary representation, realizability by an action, dual group, unitary equivalence and Kazhdan's property (T). We achieve a result regarding unitary representati- ons realizable by an action on finite abelian groups according to article [5] and show that it is possible to generalize it to all finite groups at the end of the thesis according to article [6]. A large part of the text subsequently deals with proper- ties of unitary representations and their relations. We connect the terms compact topological group and Kazhdan's property (T).

In this article the author develops a jurisprudential analysis of the right to negative freedom of association, as an essential part of the right to freedom of association, and how certain agreements, concluded within the framework of collective bargaining, contain provisions that go against the principle non-discrimination. Such agreements are declared null and void, and sometimes are admitted and incorporated into the constitutionality and legality. Based on typical cases will be discussed in which situations the agreements concluded by unions violate the right to negative freedom of association of the unaffiliated and become a discriminatory act and an unjustified pressure to join a union. / En el presente artículo se desarrolla un análisis jurisprudencial del derecho a la libertad sindical negativa, como parte esencial del derecho a la libertad sindical, y cómo ciertos acuerdos, concluidos en el marco de la negociación colectiva, contienen disposiciones que van en contra del principio de no discriminación. Dichos acuerdos son declarados nulos y sin efectos, y en otras ocasiones son admitidas e incardinados dentro de la constitucionalidad y legalidad. Sobre la base de casos paradigmáticos se expondrá en qué situaciones los acuerdos celebrados por los sindicatos atentan contra el derecho de libertad sindical negativa de los no afiliados y devienen en un acto discriminatorio y una presión injustificada a afiliarse a un sindicato.