  Kashiwara defined the Maslov index (associated to a collection of Lagrangian
subspaces of a symplectic vector space over a field F) as a class in the Witt
group W(F) of quadratic forms. We construct a canonical quadratic vector space
in this class and show how to understand the basic properties of the Maslov
index without passing to W(F)--that is, more or less, how to upgrade
Kashiwara's equalities in W(F) to canonical isomorphisms between quadratic
spaces. We also show how our canonical quadratic form occurs naturally in the
context of the Weil representation. The quadratic space is defined using
elementary linear algebra. On the other hand, it has a nice interpretation in
terms of sheaf cohomology, due to A. Beilinson.
