In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is ⁠1/2⁠n(n + 1) (where the dimension of V is 2n). It may be identified with the homogeneous space

where U(n) is the unitary group and O(n) the orthogonal group. Following Vladimir Arnold it is denoted by Λ(n). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V.

A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space V of dimension 2n. It may be identified with the homogeneous space of complex dimension ⁠1/2⁠n(n + 1)

where Sp(n) is the compact symplectic group.

To see that the Lagrangian Grassmannian Λ(n) can be identified with U(n)/O(n), note that ${\displaystyle \mathbb {C} ^{n}}$ 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 ${\displaystyle \mathbb {C} ^{n}}$ are then the real subspaces ${\displaystyle L\subseteq \mathbb {C} ^{n}}$ of real dimension n on which the imaginary part of the inner product vanishes. An example is ${\displaystyle \mathbb {R} ^{n}\subseteq \mathbb {C} ^{n}}$. The unitary group U(n) acts transitively on the set of these subspaces, and the stabilizer of ${\displaystyle \mathbb {R} ^{n}}$ is the orthogonal group ${\displaystyle \mathrm {O} (n)\subseteq \mathrm {U} (n)}$. 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).

The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: ${\displaystyle \Omega (\mathrm {Sp} /\mathrm {U} )\simeq \mathrm {U} /\mathrm {O} }$, and ${\displaystyle \Omega (\mathrm {U} /\mathrm {O} )\simeq \mathbb {Z} \times \mathrm {BO} }$ – they are thus exactly the homotopy groups of the stable orthogonal group, up to a shift in indexing (dimension).

In particular, the fundamental group of ${\displaystyle U/O}$ is infinite cyclic. Its first homology group is therefore also infinite cyclic, as is its first cohomology group, with a distinguished generator given by the square of the determinant of a unitary matrix, as a mapping to the unit circle. Arnold showed that this leads to a description of the Maslov index, introduced by V. P. Maslov.

For a Lagrangian submanifold M of V, in fact, there is a mapping