After reading chapter 9 in Lee's book on smooth manifolds, I'm trying to figure out the dimension of the Lagrangian Grassmannian in $\mathbb{R}^{2n}$, and I'm wondering if anyone can help me out.
I know that the group $Sp(2n, \mathbb R)$ acts transitively on the LG, which means that I can give it a smooth manifold structure who's dimension is equal to the codimension in $Sp(2n,\mathbb R)$ of the isotropy subgroup of an arbitrary point (so a Lagrangian plane) in the LG. But I'm not sure on how to find this codimension without resulting to some possibly complicated linear algebra.
I already know that the dimension of $Sp(2n,\mathbb R)$ is $n(2n+1)$, and I'm wondering if there's a simple way to find the dimension of this isotropy subgroup. Does anyone have an idea on how to think about this problem more simply? I'm also not completely sure on why it should be a closed subgroup, maybe knowing this could help in figuring out the dimension (maybe I can somehow write it as a level set of a smooth map?).
Thanks in advance.