nLab
Lagrangian Grassmannian
Contents
Context
Symplectic geometry
Contents
Definition
For $V$ a symplectic vector space , its Lagrangian Grassmannian $LGrass(V)$ is the space of its Lagrangian (maximal isotropic) subspaces.

Properties
As a coset space
The group of symplectomorphism s of $V$ naturally acts on $LGrass(V)$ .

The unitary group $U_J(V)$ associated to any fixed compatible complex structure $J$ on $V$ acts transitively on $LGrass(V)$ . In fact, $LGrass(V)$ is diffeomorphic to the coset space $U(n)/O(n)$ of the unitary group by the orthogonal group , where $n$ is the complex dimension of $(V,J)$ (so the real dimension of $V$ is $2n$ ).

Cohomology and Maslov index
The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers

$H^1(LGrass, \mathbb{Z})
\simeq
\mathbb{Z}
\,.$

The generator of this cohomology group is called the universal Maslov index

$u \in H^1(LGrass, \mathbb{Z})
\,.$

Given a Lagrangian submanifold $Y \hookrightarrow X$ of a symplectic manifold $(X,\omega)$ , its tangent bundle is classified by a function

$i \;\colon\; Y \to LGrass
\,.$

The Maslov index of $Y$ is the universal Maslov index pulled back along this map

$i^\ast u \in H^1(Y,\mathbb{Z})
\,.$

References
Sean Bates, Alan Weinstein , Lectures on the geometry of quantization , pdf

Andrew Ranicki , The Maslov Index (pdf )

Esteban Andruchow, Gabriel Larotonda, Lagrangian Grassmannian in Infinite Dimension (arXiv:0808.2270 )

J. Carrillo-Pacheco, F. Jarquín-Zárate, M. Velasco-Fuentes, F. Zaldívar, An explicit description in terms of Plücker coordinates of the Langrangian-Grassmannian , arxiv/1601.07501

Last revised on January 28, 2016 at 11:42:46.
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