Boundary measures for symmetric domains and integral formulas for the discrete Wallach points
The \(AA\) consider the analytic continuation of the holomorphic discrete series in the case of an irreducible hermitian symmetric domain \(B = G/K\) of rank \(r\) in \(\mathbb{C}^d\). More precisely, the discrete series of the biholomorphic automorphisms of \(G\) consists of the weighted Bergman spaces \(H_\lambda^2(B)\) of all holomorphic and square integrable (with respect to certain \(K\)-invariant measures \(\mu_\lambda\)) functions on \(B\), and the analytic continuation is with respect to the parameter \(\lambda\); two families, among the ''Wallach'' parameters are considered, namely the discrete Wallach points \(\lambda = la/s\), \(l \in \{0, \ldots, r - 1\}\), \(a\) being a numerical invariant of \(B\) and the embedded Wallach points \(\alpha_l = a/2 + la/2\) wich are not square integrable. The \(AA\) find explicit realizations of the invariant inner products for the Wallach parameters \(\lambda\), for both the above families, but which are too complicated to be given here. For the ``embedded'' W points the \(AA\) show that the invariant inner products can be realized via integration over some boundary orbits of \(G\) with respect to some semi-invariant measure; for the discrete series this is not possible, so the \(AA\) construct invariant pseudo-differential operators, which allow to obtain explicit formulae for the invariant inner products. They study the same question using the Fourier transform, in the setting of tube domains and Siegel domains, completing previous results [Doc. Math. J. DMV 2, 213--261 (1997; Zbl 0977.46010) and (Contemp. Math. 214, 1--22 (1998; Zbl 0898.47016)]. The work includes a detailed study of the boundaries of symmetric cones and Siegel domains, and of boundary integration for Cartan domains.
scientific article; zbMATH DE number 2078292
scientific article