Eigenvalues of generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains (Q730696)

The authors develop a group-theoretic method to generalize the Laplace-Beltrami operators on the classical domains. In [\textit{K. Okamoto}, Lect. Notes Math. 266, 255--271 (1972; Zbl 0234.43009)], inspired by \textit{S. Helgason}'s paper [Adv. Math. 5, 1--154 (1970; Zbl 0209.25403)], one of the authors defined the ``Poisson transforms'' for homogeneous vector bundles over symmetric spaces. In [\textit{K. Okamoto, M. Tsukamoto} and \textit{K. Yokota}, Jap. J. Math., New Ser. 26, No.~1, 51--103 (2000; Zbl 1019.32018)], the authors defined the generalized Poisson-Cauchy transforms for homogeneous holomorphic line bundles over hermitian symmetric spaces and computed explicitly the kernel functions for each type of the classical domains. In [\textit{E. Imamura, K. Okamoto, M. Tsukamoto} and \textit{A. Yamamori}, Proc. Japan Acad., Ser. A 82, No. 9, 167--172 (2006; Zbl 1154.43006)], making use of the Casimir operator, the authors defined the ``generalized Laplacians'' on homogeneous holomorphic line bundles over hermitian symmetric spaces and showed that the generalized Poisson-Cauchy transforms give rise to eigenfunctions of the ``generalized Laplacians''. In the paper under review, the authors use the canonical coordinates defined by the Harish-Chandra decomposition. This leads to a certain ``canonical Riemannian metric''. It turns out that Hua's differential operator is the Laplace-Beltrami operator defined by this metric. For each type of the classical domains, the authors carry out the direct computation to obtain the explicit formulas of line bundle valued invariant differential operators which are called the generalized Laplacians and compute their eigenvalues evaluated at the generalized Poisson-Cauchy kernel functions.