Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds (Q424761)

Let \(G\) be a connected compact Lie group, \(K\subset G\) a closed subgroup, let \(D\) be a \(G\)-invariant domain in \(X_\mathbb C:=G_\mathbb C/K_\mathbb C\). The authors prove that the identity component \(\text{Aut}(D)^0\) of the group \(\text{Aut}(D)\) of all holomorphic automorphisms of \(G\) can be realized as a closed subgroup of \(\text{Iso}(X,g)\) with a suitable \(G\)-invariant Riemannian metric \(g\) on \(X:=G/K\) whenever \(D\) is relatively compact and orbit connected. If the space \(X\) is strongly isotropy irreducible and \(D\) is relatively compact or hyperbolic and there are no copies of both the exceptional groups \(G_2\) and \(\text{Spin}(7)\) in the universal covering group of \(G\), then \(\text{Aut}(D)^0=G\). Also we have \(\text{Aut}(D)^0=G\) if \(\text{dim}(K)>0\), \(X\) is homogeneous and isotropy irreducible but not strongly isotropy irreducible and \(D\) is relatively compact or Stein \(G\)-invariant being hyperbolic in Kobayashi sense.