Looking at [this](https://mathoverflow.net/questions/262321/lagrangian-grassmannian-as-a-spin-manifold) and [this](https://mathoverflow.net/questions/247585/canonical-bundle-of-the-lagrangian-grassmannian?noredirect=1&lq=1) question about the Lagrangian Grassmannian, and its linked [Wikipedia](https://en.wikipedia.org/wiki/Lagrangian_Grassmannian) description as the quotient of $Sp(N)$ by the unitary group $U(n)$, I wondering what is the explicit embedding $U(N) \hookrightarrow Sp(2n)$. 


As far I understand, we have (also by Wikipedia) that $Sp(n)$ is the group of quaternionic unitary matrices of order $n$, and so, a subset of $M(2n,\mathbb{C})$. By analogy with the case of the usual Grassmannian, I would embed $U(n)$ in the bottom right hand corner, and put $1$'s on the remaining diagonal entries. However, it is not clear to me that this is actually contained in $Sp(n)$. Moreover, will it make a difference if I instead embed it in the top left hand corner. Because of less Dynkin diagram symmetries in the $C$-series case, I feel this should not be isomorphic. 



Explicit description of the Lagrangian Grassmannian as a homogeneous space
Like any normal person, I'm going to use the symplectic form where $\langle e\_i,e\_{j}\rangle =\pm \delta\_{j,2n-i+1}$ with $1$ if $i\leq n$ and $-1$ if $i>n$. The compact symplectic group you have in mind are the unitary matrices preserving this form. In these coordinates, you have to embed $U(n)$ as $A\mapsto \big[\begin{smallmatrix}A & 0\\ 0&A^{-1}\end{smallmatrix}\big]$; this is the subgroup of the compact symplectic group which preserves the Lagrangian subspace spanned by the vectors $e\_1,\dots, e\_n$.


