The Maslov triple index on the Shilov boundary of a classical domain

Abstract

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 ( σ 1, σ 2, σ 3)∈ S× S× S, a characteristic G-invariant ι( σ 1, σ 2, σ 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.