nLab Lagrangian Grassmannian




For VV a symplectic vector space, its Lagrangian Grassmannian LGrass(V)LGrass(V) is the space of its Lagrangian (maximal isotropic) subspaces.


As a coset space

The symplectic group of VV naturally acts on LGrass(V)LGrass(V).

The unitary group U J(V)U_J(V) associated to any fixed compatible complex structure JJ on VV acts transitively on LGrass(V)LGrass(V). In fact, LGrass(V)LGrass(V) is diffeomorphic to the coset space U(n)/O(n)U(n)/O(n) of the unitary group by the orthogonal group, where nn is the complex dimension of (V,J)(V,J) (so the real dimension of VV is 2n2n).

Cohomology and Maslov index

The first ordinary cohomology of the stable Lagrangian Grassmannian

LGrass=lim nLGrass( n) LGrass = \lim_{n \to \infty} LGrass(\mathbb{C}^n)

with integer coefficients is isomorphic to the integers

H 1(LGrass,). H^1(LGrass, \mathbb{Z}) \cong \mathbb{Z} \,.

The generator of this cohomology group is called the universal Maslov index

uH 1(LGrass,). u \in H^1(LGrass, \mathbb{Z}) \,.

Given a Lagrangian submanifold YXY \hookrightarrow X of a symplectic manifold (X,ω)(X,\omega), its tangent bundle is classified by a function

i:YLGrass. i \;\colon\; Y \to LGrass \,.

The Maslov index of YY is the universal Maslov index pulled back along this map

i *uH 1(Y,). i^\ast u \in H^1(Y,\mathbb{Z}) \,.

Unstably, a generator of

H 1(LGrass( n),)[U(n)/O(n),U(1)] H^1(\LGrass(\mathbb{C}^n),\mathbb{Z}) \cong [U(n)/O(n), U(1)]

is given by the map

det 2:U(n)U(1) \mathrm{det}^2 \colon U(n) \to U(1)

This map passes to the quotient U(n)/O(n)U(n)/O(n) because the determinant of an element of O(n)O(n) can only be ±1\pm 1.


  • 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.