The Maslov triple index on the Shilov boundary of a classical domain
Résumé
Let $D$ be an irreducible Hermitian symmetric space of tube-type, $S $its Shilov boundary, $G$ its group of holomorphic diffeomorphisms. For a generic triple of points $(\sigma_1, \sigma_2, \sigma_3) \in S \times S \times S$, a characteristic $G$-invariant $\iota (\sigma_1,\sigma_2,\sigma_3)$, called the Maslov index was introduced in [Transform. Groups 6 (2001) 303]. For $D$ of classical type (i.e. for all cases except for the exceptional domain associated to Albert's algebra), the definition of the Maslov index is extended to all triples, by using a holomorphic embedding of $D$ into a Siegel disc, which corresponds to an embedding of $S$ into a Lagrangian manifold. When $D$ is the Lie ball, the extension of the definition is obtained through a realization of $S$ in the Lagrangian manifold of a spinor space.