Differential operators on {B}ergman space on bounded symmetric domains

We classify self-adjoint first-order differential operators on weighted Bergman spaces on the N-dimensional unit ball BN and D² of 2×2 complex matrices satisfying I-ZZ^*>0.Our main tools are the discrete series representations of SU(N,1) and SU(2,2). We believe that our observations extend to general bounded symmetric domains.