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I am looking for a proof or counterexample for following assertion

Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure (In sense of Nigel Hitchin)

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In 1950s A. Borel, R. Bott, J. L. Koszul, F. Hirzebruch et al. investigated the coadjoint orbits as complex homogeneous manifolds. It was proven that each coadjoint orbit of a compact connected Lie group $G$ admits a canonical G-invariant complex structure and the only (within homotopies) $G$-invariant Kählerian metrics.

Reference: Bott R. The Geometry and Representation Theory of Compact Lie Groups, In: Representation Theory of Lie Groups, London Mathematical Society Lecture Note Series, Cambridge Univ. Press, 34 (1979) 65–90.

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  • $\begingroup$ Dietrich Burde , But what about G-invariant generalized complex structure ? this is the question $\endgroup$ – user21574 Jan 27 '14 at 14:54
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    $\begingroup$ I thought that complex and symplectic structures are two canonical examples giving generalized complex structures ? $\endgroup$ – Dietrich Burde Jan 27 '14 at 14:58

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