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.