Noyau de Cauchy-Szegö d'un espace symétrique de type Cayley. (Cauchy-Szegö kernel of a symmetric space of Cayley type.)
The author studies the geometry and harmonic analysis of certain open domains \(\Xi\) in a complex symmetric space \(G_\mathbb{C}/H_\mathbb{C}\) whose Shilov boundary is a symmetric space \(G/H\) of Cayley type. The Hardy space \(H^2 (\Xi)\) then consists of holomorphic functions on \(\Xi\) with \(L^2\)-boundary values on \(G/H\). It carries a holomorphic representation of \(G\) which decomposes discretely into representations from the holomorphic discrete series of \(G/H\) as introduced by Ørsted and Ólafsson. The main achievement of this paper is to make the decomposition of \(H^2(\Xi)\) explicit in the language of Jordan algebras, which is particularly apt to describe the Cayley type situation, and in this way obtain an explicit formula for the Cauchy-Szegö kernel. On the way the author provides several remarkable results: (i) An explicit calculation of the \(L^2\)-norm for the highest weight vectors in the irreducible components of \(H^2(\Xi)\) which as a corollary yields the spectrum of \(H^2(\Xi)\). (ii) A map \(u\mapsto\psi_u\), which associates an \(H\)-invariant distribution vector to each element of a lowest \(K\)-type of an irreducible component of \(H^2 (\Xi)\), and which is explicit enough to identify summands in the general expansion formula for the Cauchy-Szegö kernel as quotients of well studied spherical functions by certain \(c\)-functions. In the last part the author derives an isomorphism between the Hardy space \(H^2(\Xi)\) and the classical Hardy space of the domain in which \(\Xi\) is realized as an open dense subset. This isomorphism gives rise to another explicit formula for the Cauchy-Szegö kernel.
scientific article; zbMATH DE number 1202018
scientific article