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Let $G$ be a compact connected Lie group and $T$ be a maximal torus in $G$. Then the homogeneous space $G/T$ is a simply connected orientable manifold. (See, e.g., Hofmann-Morris: The structure of compact groups, page 291). I would like to know, whether $G/T$ is expressible in terms of some familiar manifolds in case when $G$ is one of the classical groups $SO(n)$, $SU(n)$, $Sp(n)$.

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As your title gives away, $G/T$ can be expressed as the manifold of full flags in $\mathbf C^n$ for $\mathrm{SU}(n)$, resp. full isotropic flags for the bilinear form defining $\mathrm{SO}(n)$ or $\mathrm{Sp}(n)$ — except that the one for $\mathrm{SO}(2l)$ splits into two open orbits. It can also be expressed as a coadjoint orbit.

All this and much more is in e.g. Alekseevskiĭ's good survey Flag manifolds (1997)(pdf). To connect with your setup, just note on p. 23 that a maximal torus $T$ is its own centralizer $Z_G(T)$.

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