Invariant metrics and invariant functions on the Reinhardt domains (Q1262980)

This brief paper records several results, mostly without proofs. Consider the Reinhardt domain \[ D(K_ 1,...,K_ p)=\{z\in {\mathbb{C}}^ n| \quad \sum^{n}_{j=1}| z_ j|^{2K_ j}<1\} \] where \(K_ 1\geq...\geq K_ p>1\), \(K_{p+1}=...=K_ n=1\). When \(p=1\) write \(D(K_ 1)=D(K)\). The author lists the automorphisms of \(D(K_ 1,...,K_ p)\) in case \(K_ 1,...,K_ p\) are all different, and recalls a formula of D'Angelo for the Bergmann kernel function of D(K). The latter formula leads to the construction of a family of invariant Kähler metrics and of an invariant function on D(K). The author asks whether this invariant function generates all such functions.