To see that the Lagrangian Grassmannian Λ(n) can be identified with U(n)/O(n), note that is a 2n-dimensional real vector space, with the imaginary part of its usual inner product making it into a symplectic vector space. The Lagrangian subspaces of are then the real subspaces of real dimension n on which the imaginary part of the inner product vanishes. An example is . The unitary groupU(n) acts transitively on the set of these subspaces, and the stabilizer of is the orthogonal group. It follows from the theory of homogeneous spaces that Λ(n) is isomorphic to U(n)/O(n) as a homogeneous space of U(n).
Topology
The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: , and – they are thus exactly the homotopy groups of the stable orthogonal group, up to a shift in indexing (dimension).
which classifies its tangent space at each point (cf. Gauss map). The Maslov index is the pullback via this mapping, in
of the distinguished generator of
.
Maslov index
A path of symplectomorphisms of a symplectic vector space may be assigned a Maslov index, named after V. P. Maslov; it will be an integer if the path is a loop, and a half-integer in general.
It appeared originally in the study of the WKB approximation and appears frequently in the study of quantization, quantum chaos trace formulas, and in symplectic geometry and topology. It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds.
References
^Wendl, Chris. "Lectures on Symplectic Field Theory". arXiv:1612.01009.
V. I. Arnold, Characteristic class entering in quantization conditions, Funktsional'nyi Analiz i Ego Prilozheniya, 1967, 1,1, 1-14, doi:10.1007/BF01075861.
V. P. Maslov, Théorie des perturbations et méthodes asymptotiques. 1972
Ranicki, Andrew, The Maslov index home page, archived from the original on 2015-12-01, retrieved 2009-10-23 Assorted source material relating to the Maslov index.