  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.
