## Hiroshima Mathematical Journal

### Eigenvalues of generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains

#### Abstract

We develop a group-theoretic method to generalize the Laplace-Beltrami operators on the classical domains. In K. Okamoto, "Harmonic analysis on homogeneous vector bundles," Lecture Notes in Mathematics, Springer-Verlag, 266 (1971), 255–271, inspired by Helgason's paper, "A duality for symmetric spaces with applications to group representations," Advan. Math. 5 (1970), 1–154, we defined the "Poisson transforms" for homogeneous vector bundles over symmetric spaces. In K. Okamoto, M. Tsukamoto and K. Yokota, "Generalized Poisson and Cauchy kernel functions on classical domains," Japan. J. Math. 26 No. 1 (2000), 51–103., we 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 E. Imamura, K. Okamoto, M. Tsukamoto and A. Yamamori, "Generalized Laplacians for Generalized Poisson-Cauchy transforms on classical domains," Proc. Japan Acad., 82, Ser. A (2006), 167–172., making use of the Casimir operator, we 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 this paper, using the canonical coordinates for each type of the classical domains, we carry out the direct computation to obtain the explicit formulas of (line bundle valued) invariant differential operators which we call the generalized Laplacians and compute their eigenvalues evaluated at the generalized Poisson-Cauchy kernel functions

#### Article information

Source
Hiroshima Math. J., Volume 39, Number 2 (2009), 237-275.

Dates
First available in Project Euclid: 31 July 2009

https://projecteuclid.org/euclid.hmj/1249046339

Digital Object Identifier
doi:10.32917/hmj/1249046339

Mathematical Reviews number (MathSciNet)
MR2543652

Zentralblatt MATH identifier
1179.43008

#### Citation

Imamura, Eisuke; Okamoto, Kiyosato; Tsukamoto, Michiroh; Yamamori, Atsushi. Eigenvalues of generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains. Hiroshima Math. J. 39 (2009), no. 2, 237--275. doi:10.32917/hmj/1249046339. https://projecteuclid.org/euclid.hmj/1249046339