The Maslov index as a quadratic space

Abstract

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 canon- ical isomorphisms between quadratic spaces. The quadratic space is defined using ele- mentary linear algebra. On the other hand, it has a nice interpretation in terms of sheaf cohomology, due to A. Beilinson.