Hallo,


Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G\_{\mathbb{C}}$. Does there exist a Kähler structure on $G\_{\mathbb{C}}$ which is $G$-invariant (maybe in a neighbourhood of $G$ in $G\_{\mathbb{C}}$)?


hapchiu



Kähler form on complex Lie group
Yes, such a Kähler form always exists: Embed $G$ as a matrix group in $\mathrm{SU}(n)$ for some $n$ and then let $G\_\mathbb{C}\subset \mathrm{SL}(n,\mathbb{C})\subset M\_n(\mathbb{C})$ be the complexification. Choose a Kähler form on this latter vector space, pull it back to $G\_\mathbb{C}$ and then, using the compactness of $G$, average its pullbacks under left and right multiplications. This will yield a Kähler form on $G\_\mathbb{C}$ that is $G$-invariant.


