We know that $U(N)$ can be embedded into $SU(n+1)$ and that the quotient is isomorphic to complex projective space: $$ SU(n+1)/U(n) \simeq {\mathbb CP}^{n}. $$ We can split this process into two stages, first quotient $SU(n+1)$ by $SU(n)$ giving the sphere $$ SU(n+1)/SU(n) \simeq S^{2n+1}, $$ and then quotient $S^{2+1}$ by $U(1)$ to get ${\mathbb CP}^{n}$.

I am curious about the analogue for the compact symplectic groups $Sp(n)$. According to the Hermitian Symmetric Space Wikipedia page, $Sp(n)/U(n)$ is the space of complex structures on $\mathbb{H}^n$ compatible with the inner product. If it helps, note that like ${\mathbb CP}^{n}$, this is an Hermitian Symmetric Space, and a flag manifold. One can again split this quotient into a quotient by $SU(N)$, and a quotient by $U(1)$.

**Question: What is the space $Sp(n)/SU(N)$? Does it have a name?**