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Let E be a real symplectic vector space. Choose a compatible complex structure so that E is the real vector space underlying a Hermitian vector space W. Let Lag(E) denote the Grassmann manifold of Lagrangian subspaces L ⊂ E. The inclusion induces an isomorphism L ⊗ C ∼ = W for any Lagrangian L. Fix a complex volume form Φ ⊂ Det W ∗ of unit norm. There are two points in Det L ⊂ Det L ⊗ C of unit norm: evaluate Φ on each and multiply the result. This produces a function Lag(E) → T to the circle whose homotopy class is a generator of H 1 (Lag(E); Z) called the Maslov class. This construction shows that its mod 2 reduction is represented by the determinant line bundle of the tautological vector bundle over Lag(E). Its mod 4 reduction appears in microlocal analysis. In this article Weinstein considers the complex analog of this construction. Thus let E be a complex symplectic vector space, that is, a complex vector space (necessarily of even complex dimension) equipped with a nondegenerate skew-symmetric complex-valued bilinear form. Let η → Lag(E) be the tautological vector bundle over the Lagrangian Grassmannian and Det(η ∗ ) the determinant line bundle of its dual. The latter has a characteristic class in H 2 (Lag(E); Z) which is the complex analog of the Maslov class. Its reduction mod 2, the obstruction to finding a squar

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Year: 2013

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