Primitive du cocycle de Maslov g\'en\'eralis\'e

Let $\mathcal{D}$ be a Hermitian symmetric space of tube type, and let $S$ be its Shilov boundary. We give a realization of the universal covering $\widetilde{S}$ of $S$. Then we describe on $\widetilde{S}$ a primitive for the generalized Maslov cocycle as defined in [{\it Transform. Groups} {\bf 6} (2001), 303-320] and [{\it J. Math. Pures Appl.} {\bf 83} (2004), 99-114]. It generalizes the Souriau index in the case of the Lagrangian manifold. A variation of this construction yields a generalization of the Arnold-Leray-Maslov index. This primitive is used to generalize the symplectic rotation number.