Isotropic Grassmannian - Imported Knowlpedia Content

Definition

Let $(V,\omega)$ be a $2n$ -dimensional real or complex symplectic vector space, and let $0\leq k\leq n$ . The isotropic Grassmannian is

\operatorname{IGr}_k(V,\omega) =\{L\in\operatorname{Gr}_k(V):\omega|_{L\times L}=0\}.

It is the subspace of the Grassmannian consisting of $k$ -planes that are isotropic. The bound $k\leq n$ is forced by nondegeneracy of $\omega$ . Over $\mathbb R$ it is a smooth compact manifold; over $\mathbb C$ it is also a smooth projective variety. Its points therefore parametrize isotropic subspaces while retaining their position inside the fixed ambient symplectic space.

Homogeneous-space structure

The symplectic group of $(V,\omega)$ acts transitively on $\operatorname{IGr}_k(V,\omega)$ . Over $\mathbb C$ , the stabilizer of an isotropic $k$ -plane is a parabolic subgroup, so the isotropic Grassmannian is a homogeneous projective variety. Its dimension over the base field is

k(2n-k)-\binom{k}{2}.

This results from imposing the $\binom{k}{2}$ independent equations expressing the vanishing of $\omega$ on the tautological $k$ -plane Fulton–Harris, §17.3.

Extremal cases and examples

For $k=1$ , every line is isotropic, so $\operatorname{IGr}_1(V,\omega)=\operatorname{Gr}_1(V)$ . For $k=n$ , isotropic subspaces are maximal and therefore Lagrangian; the resulting space is the Lagrangian Grassmannian. A $k$ -plane on which $\omega$ has nonzero restriction is a decisive non-example.

Conventions and scope

“Isotropic Grassmannian” is also used for quadratic spaces, where isotropy is defined by a symmetric quadratic form and the resulting orthogonal Grassmannian has different geometry. The notation $\operatorname{IGr}$ here always refers to a nondegenerate alternating form. The real smooth manifold and complex projective variety are two scalar-field realizations of the same incidence condition.

References

William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991. Springer DOI record. Relevant: §17.3, symplectic groups, isotropic flags, and their homogeneous spaces.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2001. Springer DOI record. Relevant: Chapter 1, symplectic linear algebra and isotropic and Lagrangian subspaces.