# Subvarieties of Lagrangian Grassmannians

Let $$LG(n,2n)$$ be the Lagrangian Grassmannian parametrizing Lagrangian subspaces (so of dimension $$n$$) of $$\mathbb{C}^{2n}$$. Then $$LG(n,2n)\subset G(n,2n)$$, where $$G(n,2n)$$ is the Grassmannian of subspaces of dimension $$n$$ of $$\mathbb{C}^{2n}$$.

Fix a subspace $$H\subset \mathbb{C}^{2n}$$ of dimension $$n+2$$, and denote by $$G(n,H)\subset G(n,2n)$$ the Grassmannian of subspaces of dimension $$n$$ that are contained in $$H$$.

The intersection $$X_n := LG(n,2n)\cap G(n,H)\subset G(n,2n)$$ parametrizes Lagrangian subspaces of $$\mathbb{C}^{2n}$$ that are contained in $$H$$.

For instance, for $$n = 2$$ we have $$G(2,H) = G(2,4)$$ and hence $$X_2 = LG(2,4)$$.

In general, is the variety $$X_n$$ smooth and irreducible? By any chance is $$X_n$$ a well-known variety appearing under some name in the literature?

Addition: Is the subvariety $$Y_n\subset G(n+2,2n)$$, parametrizing $$(n+2)$$-dimensional subspaces $$H\subset \mathbb{C}^{2n}$$ that are co-isotropic, homogeneous as well? Is there a formula for the dimension of $$Y_n$$?

Thank you very much.

Note first that, if $$L$$ is a Lagrangian contained in $$H$$ then $$L^\perp = L$$ contains $$H^\perp$$. So $$X_n$$ is non-empty only when $$H$$ is co-isotropic for your symplectic form.

When $$H$$ is co-isotropic, the symplectic form $$\omega$$ induces a symplectic form on $$H/H^\perp$$ (which has dimension $$4$$) and Lagrangians of $$\mathbb C^{2n}$$ contained in $$H$$ are in bijection with Lagrangian subspaces of $$H/H^\perp$$ (since they all contain $$H^\perp$$). Thus $$X_n$$ is a copy of $$LG(2,4)$$ inside $$LG(n,2n)$$. It is homogeneous under the parabolic subgroup of $$Sp(2n,\mathbb C)$$ preserving $$H$$, so it is smooth and irreducible.

• Thank you for the answer. Is the subvariety $Y_n\subset G(n+2,2n)$, parametrizing $(n+2)$-dimensional subspaces $H\subset\mathbb{C}^{2n}$ that are co-isotropic, homogeneous as well? Is there a formula for the dimension of $Y_n$? Commented Jun 7, 2021 at 22:11

Co-isotropic subspaces of dimension $$n+k$$ are in bijection with isotropic subspaces of dimension $$n-k$$.

The variety of isotropic subspaces of dimension $$n-k$$ of a symplectic vector space of dimension $$2n$$ is the symplectic Grassmannian $$SG(n-k,2n)$$. Its dimension is given by

$$\dim(SG(n-k,2n)) = 2n(n-k)-\frac{3(n-k)^2-(n-k)}{2}.$$

In particular, when $$k = 0$$ you get the Lagrangian Grassmannian $$LG(n,2n)$$ parameterizing maximal isotropic subspaces, and its dimension is given by $$\dim(LG(n,2n)) = \frac{n(n+1)}{2}.$$