For a symplectic vector space, its Lagrangian Grassmannian is the space of its Lagrangian (maximal isotropic) subspaces.
The symplectic group of naturally acts on .
The unitary group associated to any fixed compatible complex structure on acts transitively on . In fact, is diffeomorphic to the coset space of the unitary group by the orthogonal group, where is the complex dimension of (so the real dimension of is ).
The first ordinary cohomology of the stable Lagrangian Grassmannian
with integer coefficients is isomorphic to the integers
The generator of this cohomology group is called the universal Maslov index
Given a Lagrangian submanifold of a symplectic manifold , its tangent bundle is classified by a function
The Maslov index of is the universal Maslov index pulled back along this map
Unstably, a generator of
is given by the map
This map passes to the quotient because the determinant of an element of can only be .
Lagrangian Grassmannian
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 18, 2023 at 23:28:17. See the history of this page for a list of all contributions to it.