# Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$

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)

• 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. – Alexander Chervov Jan 9 '14 at 17:07
• Dear Alexander Chervov, can you write your email for me? – user21574 Jan 9 '14 at 17:13
• 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 – user21574 Jan 9 '14 at 17:17
• Al.mylastname at gmail.com Omega ( x y ) is anti symmetric is not it ? – Alexander Chervov Jan 9 '14 at 18:23
• Thanks for email. $\omega(X,\bar Y)$ is symplectic fors and the second equality is KKS symplectic form – user21574 Jan 9 '14 at 18:46