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Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$.(note that $h=g-i\omega$ and for $\omega$ we have Kirilov symplectic 2-form )(maybe it be possible by Hodge star , but i couldn't find)

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    $\begingroup$ There are Kahler and hyper-Kaehler metrics on orbits of real and complex semisimple Lie algebras, as far as I understand the explicit description is not known. Hyper-Kaehler metric contructed by Kronheimer and related to Nahm monopole equations. $\endgroup$ – Alexander Chervov Jan 9 '14 at 17:07
  • $\begingroup$ Dear Alexander Chervov, can you write your email for me? $\endgroup$ – user21574 Jan 9 '14 at 17:13
  • $\begingroup$ What about if we define the hermitian form as $h(X,Y)=-i\omega(X,\bar Y)=-i<F,[X,\bar Y]>$, but I am still looking for explicit description and maybe by Hodge star $\endgroup$ – user21574 Jan 9 '14 at 17:17
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    $\begingroup$ Al.mylastname at gmail.com Omega ( x y ) is anti symmetric is not it ? $\endgroup$ – Alexander Chervov Jan 9 '14 at 18:23
  • $\begingroup$ Thanks for email. $\omega(X,\bar Y) $ is symplectic fors and the second equality is KKS symplectic form $\endgroup$ – user21574 Jan 9 '14 at 18:46

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