the Lagrangian Grassmannian

Inside a single symplectic vector space — the model phase space (R^{2n}, omega_0) — there are many Lagrangian subspaces: the q-plane, the p-plane, and infinitely many tilted half-dimensional planes on which omega vanishes. The Lagrangian Grassmannian is the space of all of them at once, packaged as a smooth manifold in its own right. It is the linear-algebra stage on which the Maslov index is defined, the way a moving Lagrangian (the tangent planes along a path of submanifolds) winds around.

Precisely, fix a symplectic vector space (V, omega) of dimension 2n. The Lagrangian Grassmannian Lambda(n) is the set of all Lagrangian subspaces L of V (n-dimensional subspaces with omega|_L = 0), given its natural smooth structure. The unitary group acts transitively on it: identifying V with C^n, Lambda(n) is the homogeneous space U(n)/O(n), a compact manifold of dimension n(n+1)/2. Its single most important topological feature is its fundamental group: pi_1(Lambda(n)) = Z, with the isomorphism given by the square of the determinant map L -> det(U)^2 in C (a Lagrangian L = U . R^n picks out U up to O(n), and det^2 kills the O(n) ambiguity). That Z-valued winding number is precisely what the Maslov index counts.

Why it matters: many constructions in symplectic and contact geometry — the Maslov class of a Lagrangian, the grading in Floer homology, semiclassical phase corrections in WKB and the Maslov correction to Bohr-Sommerfeld quantization — all live on this fundamental group pi_1(Lambda(n)) = Z. The Lagrangian Grassmannian is the universal local model: along any path of Lagrangian planes (for instance the tangent spaces to a Lagrangian submanifold moving in a family, or the linearized flow of a Hamiltonian system) you get a loop or path in Lambda(n), and its homotopy data is the invariant. A point to keep straight: this is NOT the ordinary Grassmannian of n-planes in R^{2n}; the Lagrangian condition omega|_L = 0 cuts it down to a much smaller, and topologically very different, subspace.