Bergman and Szegö kernels for strictly pseudo-convex domains which generalize the unit ball. (Q1199339)

Let \(G\) be a complex semi-simple Lie group and \(K\) a maximal compact subgroup of \(G\). Let \(\Lambda\) be a dominant weight of \(G\) and \((\pi_ \Lambda,E_ \Lambda)\) be the irreducible representation of \(G\) associated to the dominant weight \(\Lambda\). We fix a \(K\)-invariant hermitian scalar product \((\;|\;)\) on \(E_ \Lambda\). Let \(\Omega\) be the intersection of the unit ball in \(E_ \Lambda\) with the orbit \(G.v_ \Lambda\), where \(v_ \Lambda\) is the dominant vector. In the case \(G=SL(n,\mathbb{C})\), \(E_ \Lambda=\mathbb{C}^ n\) with usual scalar product \((\;|\;)\) on \(\mathbb{C}^ n\), \(\Omega\) becomes the usual ball of \(\mathbb{C}^ n\). Generalizing the above special case, the author proves that the Bergman (resp. Szegö) kernel of \(\Omega\) is a rational fraction of \((\pi_ \Lambda(g1)v_ \Lambda\mid\pi_ \Lambda(g2)v_ \Lambda)\) \((g1,g2\in G)\) and can be expressed in terms of some invariants of \(G\) and \(\pi_ \Lambda\).