Lagrangian Grassmannian

Fix complementary Lagrangian subspaces $V=L\oplus L'$. Every Lagrangian plane transverse to $L'$ is the graph of a linear map $A:L\to L'$. Identifying $L'\cong L^*$ using $\omega$, the graph is Lagrangian exactly when the associated bilinear form is symmetric. Thus this chart is modeled on $\operatorname{Sym}^2(L^*)$, and

The homogeneous-space description shows that $\Lambda(V)$ is compact and connected. Its fundamental group is infinite cyclic. A preferred generator of $H^1(\Lambda(V);\mathbb Z)$ is the universal Maslov class; pulling it back along a loop or a Lagrangian Gauss map produces the corresponding Maslov invariant. Arnol'd relates this class to the cycle of planes meeting a fixed Lagrangian nontrivially Arnol'd, 1967.

For $n=1$, every line in $\mathbb R^2$ is Lagrangian, so $\Lambda(1)=\mathbb RP^1\cong S^1$. For $n>1$, the Lagrangian Grassmannian is a proper submanifold of $\operatorname{Gr}_n(\mathbb R^{2n})$: an $n$-plane on which $\omega$ does not vanish is not a point of $\Lambda(V)$.

The notation $\Lambda(n)$ means the Lagrangian Grassmannian of the standard symplectic $\mathbb R^{2n}$. The identification with $U(n)/O(n)$ depends on auxiliary compatible linear data, although its diffeomorphism type does not. The complex Lagrangian Grassmannian of a complex symplectic vector space is a different homogeneous variety and should not be conflated with this real manifold.

1. V. I. Arnol'd, “On a characteristic class entering into conditions of quantization,” Functional Analysis and Its Applications 1 (1967), 1–14. DOI record. Relevant: the Lagrangian Grassmannian, Maslov cycle, and characteristic class.
2. Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: §2.1, linear symplectic geometry and Lagrangian subspaces.