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 $2\times 2$ complex matrices satisfying $I-Z{Z}^{\ast }>0$.Our main tools are the discrete series representations of $\mathrm{S}\mathrm{U}(N,1)$ and $\mathrm{S}\mathrm{U}(2,2)$. We believe that our observations extend to general bounded symmetric domains.