L'indice de Maslov généralisé

Abstract

Let D be a Hermitian symmetric space of tube type, G=G( D) its group of holomorphic diffeomorphisms, and S its Shilov boundary. To any triple ( σ 1, σ 2, σ 3)∈ S× S× S is associated an integer ι( σ 1, σ 2, σ 3), called its Maslov index. The Maslov index is invariant under the action of G, is skew-symmetric with respect to the three arguments and satisfies a cocycle relation. It generalizes the classical theory of the Maslov index, where S is the Lagrangian manifold and G the symplectic group. The definition of the Maslov index follows previous work [Clerc, Ørsted, Transformation Groups 6 (2001) 303–320; Clerc, Ørsted, Asian J. Math. 7 (2003) 269–296], where the definition was restricted to mutually transverse triples. The key to the present extension is the use of Γ-radial convergence at a point of the Shilov boundary.