The Maslov index revisited

Abstract

LetD be a Hermitian symmetric space of tube type,S its Shilov boundary andG the neutral component of the group of bi-holomorphic diffeomorphisms ofD. In the model situationD is the Siegel disc,S is the manifold of Lagrangian subspaces andG is the symplectic group. We introduce a notion of transversality for pairs of elements inS, and then study the action ofG on the set of triples of mutually transversal points inS. We show that there is a finite number ofG-orbits, and to each orbit we associate an integer, thus generalizing theMaslov index. Using the scalar automorphy kernel ofD, we construct a ℂ*,G-invariant kernel onD×D×D. Taking a specific determination of its argument and studying its limit when approaching the Shilov boundary, we are able to define a ℤ-valued,G-invariant kernel for triples of mutually transversal points inS. It is shown to coincide with the Maslov index. Symmetry properties and cocycle properties of the Maslov index are then easily obtained.