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.