In this paper by Onofri, they mention an explicit formula (eq. 13) for calculating a Lie algebra representation on the space of coherent states.
$$ [\hat{X}\psi](z) = (H(z,\bar{z}) + iXf(z,\bar{z}))\psi(z) - i(X\psi)(z) $$
where $H(z,\bar{z}) = \langle z|\hat{X}|z\rangle/\langle z|z\rangle$ is the Berezin symbol, $f(z,\bar{z}) = \log|\langle0|z\rangle|^{-2}$ is the Kaehler potential, and the vector field $X$ is given by $i_X\omega = dH$, where $\omega = i\partial\bar{\partial}f$ is the Kaehler 2-form.
While I've been able to apply this formula somewhat successfully to $\mathfrak{su}(1,1)$ coherent states, I have no clue where it actually comes from, which makes it rather difficult and frustrating to use, since I can't rederive it. How exactly does the justification/derivation of this formula work? Would greatly appreciate some suggestions!