The Maslov index as a quadratic space

Details The Maslov index as a quadratic space The Maslov index as a quadratic space

2005-05-26

arxiv.org/abs/math/0505561v3

Math. Res. Lett. vol 13 no 6 (2006), 985--999

Mathematics

Symplectic Geometry

20 pages, 2 figures. Presumably final version. The published version omits sections 9-11

Scientific paper

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.

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