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](https://projecteuclid.org/download/pdf_1/euclid.ojm/1353054814).


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.



Topology/geometry of $O(2n)/U(n)$
You may be remembering papers by Vogan ([1987](//ams.org/mathscinet-getitem?mr=89h:22034), p. 262; [2008](//ams.org/mathscinet-getitem?mr=2009f:22013), prop. [6.9](http://math.mit.edu/~dav/paper.html)). 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 $\mathrm O(2n)$.


