M ar 2 00 6 Invariant Differential Operators and Characters of the AdS 4 Algebra

The aim of this paper is to apply systematically to AdS 4 some modern tools in the representation theory of Lie algebras which are easily generalised to the supersymmetric and quantum group settings and necessary for applications to string theory and integrable models. Here we introduce the necessary representations of the AdS 4 algebra and group. We give explicitly all singular (null) vectors of the reducible AdS 4 Verma modules. These are used to obtain the AdS 4 invariant differential operators. Using this we display a new structure a diagram involving four partially equivalent reducible representations one of which contains all finite-dimensional irreps of the AdS 4 algebra. We study in more detail the cases involving UIRs, in particular, the Di and the Rac singletons, and the massless UIRs. In the massless case we discover the structure of sets of 2s 0 − 1 conserved currents for each spin s 0 UIR, s 0 = 1, 3 2 ,. .. All massless cases are contained in a one-parameter subfamily of the quartet diagrams mentioned above, the parameter being the spin s 0. Further we give the classification of the so(5, C I) irreps presented in a diagrammatic way which makes easy the derivation of all character formulae. The paper concludes with a speculation on the possible applications of the character formulae to integrable models.