Differential systems of type (1,1) on Hermitian symmetric spaces and their solutions

This paper concerns G-invariant systems of second-order differential operators on irreducible Hermitian symmetric spaces G/K. The systems of type (1,1) are obtained from K-invariant subspaces of p+⊗p−. We show that all such systems can be derived from a decomposition p+⊗p−=H′⊕L⊕Hc. Here L gives the Laplace–Beltrami operator and H=H′⊕L is the celebrated Hua system, which has been extensively studied elsewhere. Our main result asserts that for G/K of rank at least two, a bounded real-valued function is annihilated by the system L⊕Hc if and only if it is the real part of a holomorphic function. In view of previous work, one obtains a complete characterization of the bounded functions that are solutions for any system of type (1,1) which contains the Laplace–Beltrami operator.