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# Mathematical Research Letters

## Volume 13 (2006)

### Number 6

### The Maslov index as a quadratic space

Pages: 985 – 999

DOI: http://dx.doi.org/10.4310/MRL.2006.v13.n6.a13

#### Author

#### 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 canonical isomorphisms between quadratic spaces. 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.