I'm learning about Maass--Shimura operators, and there's a term that I'm not sure how to generalize nicely.

Let $\mathfrak{h}$ be the upper half-plane with parameter $z= x + iy$, and write $s = \frac{1}{z - \overline{z}} = \frac{1}{2iy}$. For a modular form $f \colon \mathfrak{h} \to \mathbb{C}$ of weight $k$, the Maass--Shimura operator is $D_k(f) = s^k \frac{\partial}{\partial z}\left[ s^{-k}f \right] = s^k \frac{\partial}{\partial z}[s^{-k}] f + s^ks^{-k}\frac{\partial f}{\partial z} = ksf + \frac{\partial f}{\partial z}$. The $k$ can be viewed as coming from the action of an element of the Lie algebra of $\mathbb{G}_m$. The $s$ pops out if you do the calculation directly, since you lose a power of $y$. This is a pretty nice, simple formula.

On the other hand, let $\mathfrak{h}_n = \{Z \in M_n(\mathbb{C}) \mid \, ^t\!\!Z = Z, Z = X + iY \text{ with }Y \text{ positive definite}\}$ and write $s = (Z - \overline{Z})^{-1}$. Fix a representation $\rho \colon \operatorname{GL}_n(\mathbb{C}) \to \operatorname{GL}(V)$. For a Siegel modular form of genus $n$ and weight $\rho$, $f \colon \mathfrak{h}_n \to \mathbb{C}$, the Maass-Shimura operator is $D_\rho(f) = \rho(s)\operatorname{d}\left[ \rho(s^{-1})f \right]$, where $\operatorname{d}$ is the usual exterior derivative. Using the product rule, we should get a term which is $\operatorname{d}(f)$, and a term which is $\rho(s) \operatorname{d}(\rho(s^{-1}))$ times $f$. I don't know how to calculate this, at least to the point of getting a nice, simple formula like the $skf$ above.

In fact, I am interested in the directional derivative $\rho(Z - \overline{Z})^{-1}\frac{\partial \rho(Z - \overline{Z})}{\partial z_{ij}}$, where the $z_{ij}$ means the partial derivative with respect to the $ij$ entry of the matrix $Z$. It should be able to be given in terms of the action of the Lie algebra of $\operatorname{GL}_n(\mathbb{C})$.

Maybe a general formulation of the question: Let $s \colon \mathfrak{H} \to G(\mathbb{C})$ be a map from a complex manifold $\mathfrak{H}$ to the complex points of an algebraic group $G$. Then let $\rho \colon G(\mathbb{C}) \to \operatorname{GL}(V)$ be a representation of $G$ on the complex vector space $V$. Let $\frac{\partial}{\partial z_{ij}} \in T_{\mathfrak{H}}(U)$ be a vector field on $\mathfrak{H}$. How do I find $\rho(s(z))\frac{\partial \rho(s(z)^{-1})}{\partial z_{ij}}$ in terms of the action of the Lie algebra?