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 $\mathbb{B}^N$ and $\mathbb{D}^2$ of $2\times2$ complex matrices satisfying $I-ZZ^*>0$.Our main tools are the discrete series representations of $\mathrm{SU}(N,1)$ and $\mathrm{SU}(2,2)$. We believe that our observations extend to general bounded symmetric domains.