A note on homogeneous Kähler manifolds of semi-simple Lie groups (Q1894179)

In this short note the author proves the following Theorem. Let \(G/K\) be a homogeneous Kähler manifold of a semi-simple Lie group \(G\). Then \(\text{Aut} (G/K)\) is semi-simple. Assume further that \(G\) acts effectively on \(G/K\) and that every simple factor of \(G\) is of non- compact type. Then \(G\) coincides with \(\text{Aut} (G/K)_0\). Here \(\text{Aut}(G/K)\) denotes the group of all holomorphic isometries of \(G/K\) and \(\text{Aut} (G/K)_0\) its identity component. The theorem is generalising the well-known fact that for a Hermitian symmetric space of non-compact type \(G/K\), \(G\) coincides with \(\text{Aut}(G/K)_0\) under the effectiveness assumption on \(G\).