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For any sequence of integers $0<n_1<...<n_k$, there is a flag manifold of type $(n_1, ..., n_k)$, which is the collection of ordered sets of vector subspaces of $R^{(n_k)}$ $(V_1, ..., V_k)$ with dim$(V_i)=n_i$ and $V_i$ is a subspace of $V_{(i+1)}$. There are also complex flag manifolds with complex subspaces of $C^{(n_k)}$ instead of real subspaces of a real $n_k$-space.

My questions: One particular case of flag manifolds of are $$ \frac{U(N)}{U(N_1) \times U(N_2) \times \dots U(N_M)} $$ where $\sum_{i=1}^{M}N_i=N$.

  1. What is the full isometric group of this flag manifold $\frac{U(N)}{U(N_1) \times U(N_2) \times \dots U(N_M)}? $

  2. In the special case of a complex projective space, $ \frac{U(N)}{U(1) \times U(N-1)}=\mathbb{CP}^{N-1} $, what is the full isometric group?

For both questions, I would like to know the Refs for the full isometric group that also contains not only the orientation preserving map, but also the orientation reversal map.

Partial answers/Refs are still welcome! Thanks!

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    $\begingroup$ Which metric(s) are you talking about? $\endgroup$
    – R W
    Commented Oct 24, 2018 at 2:24
  • $\begingroup$ @R W, I am guessing wonderich is asking the symmetry group of the manifolds. Does it matter which metrics you choose? $\endgroup$
    – annie marie cœur
    Commented Oct 24, 2018 at 16:04
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    $\begingroup$ If you consider them as complex algebro-geometric varieties, the answer (computed in full generality by Demazure in eudml.org/doc/142464 ) is either $PGL_n({\mathbb C})$ or the semi-direct product of the latter with ${\mathbb Z}/2$ (acting as the contragradient map). Then you can either mimic the proof in the category of manifolds or use the main principle of GAGA by Serre (the latter, however, will give only the holomorphic isometries). $\endgroup$
    – Victor Petrov
    Commented Oct 24, 2018 at 17:07

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