This is a very good paper that will be instrumental in disseminating the symplectic credo among geometers and analysts who are (still) nonbelievers. First, the authors give four different definitions for the Maslov index; the two geometric ones use intersection numbers, while the analytic ones are based on an eta invariant of an associated self-adjoint operator. In passing, it is shown that certain properness assumptions usually taken are unnecessary.
Higher indices are also discussed. The triple index is shown to be equivalent to Kashiwara-Wall’s index. On another token, Duistermaat’s single index also can be expressed in terms of Maslov indices.
The link between pairs of Lagrangian subspaces and self-adjoint operators is obtained with the introduction of a complex structure together with an Hermitian inner product. Useful relationships between the geometric and analytic constructions are obtained via the Atiyah-Patodi-Singer index theorem.
In the final sections, other perspectives are discussed, e.g., work by Floer, Salamon/Zehnder, Mrowka, Nicolaesco, and several others. The bibliography ranges from the basic to the recent work on the area. It is observed by the authors that the ideas giving rise to Maslov index first appeared in the work of J. B. Keller on semiclassical quantization; nowadays it became an important tool for geometry and analysis.