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

Abstract 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 ⊕ H c . 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 ⊕ H c 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.