Topology/geometry of $O(2n)/U(n)$

Question

I am interested in what is known about the topology (diffeomorphism type) or geometry (is it complex? non-complex? symplectic?) of either the compact space of orthonormal complex structures on $\mathbb{R}^{2n}$, $O(2n)/U(n)$, or the full space of complex structures, $Gl(2n,\mathbb{R})/Gl(n,\mathbb{C})$ (which retracts onto the compact one). I know that the algebraic topology and CW-structure of the former was worked out by Yokota.

I think I stumbled upon a $Gl(2n,\mathbb{R})$-invariant (integrable) complex structure on the non-compact quotient, and I feel it already is somewhere in the literature (hence the tag), but a brief search didn't turn out anything. Maybe these manifolds have a specific name and/or are studied extensively in a specific subfield? Any pointers will be greatly appreciated.

Answer 1

You may be remembering papers by Vogan (1987, p. 262; 2008, prop. 6.9). There he describes:

(a) $\mathrm{GL}(2n,\mathbf R)/\mathrm{GL}(n,\mathbf C)\cong\{\!$complex structures on $\mathbf R^{2n}\}$: an elliptic coadjoint orbit of $\mathrm{GL}(2n,\mathbf R)$, hence symplectic and pseudo-kähler — with signature $\left({\frac12}(n^2-n),{\frac12}(n^2+n)\right)$ (over $\mathbf C$).

(b) $\mathrm O(2n)/\mathrm U(n)\cong\{\!$orthogonal complex structures on $\mathbf R^{2n}\}=$ the $\mathrm O(2n)$-orbit of $\bigl(\begin{smallmatrix}0&-I\\I&\phantom{-}0\end{smallmatrix}\bigr)$ in (a): a complex submanifold with signature $\left({\frac12}(n^2-n),0\right)$, hence symplectic and a (Kähler) coadjoint orbit of $\mathrm O(2n)$.

Answer 2

Well, $\mathrm{SO}(2n)/\mathrm{U}(n)$ is a well-known Hermitian symmetric space, DIII. In particular, it is a compact complex manifold. See, for example, Helgason's treatment in his Differential Geometry, Lie Groups, and Symmetric Spaces.