Invariant differential operators on Hermitian symmetric spaces and their eigenvalues

Abstract

Let\(\bar D\) be the invariant Cauchy Riemann operator and\(\mathcal{M}_m = D^m \bar D^m \) the corresponding invariant Laplacians on a bounded symmetric domain. We calculate the eigenvalues ofM m on spherical functions. In particular we prove that for a symmetric domain of rank two the operatorsM 1,M 3 generate all invariant differential operators. We also find the eigenvalues of the generators introduced by Shimura.