Derivative of a representation

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?

EDIT: I should include the fact that I know what the answer should be, from the paper I'm reading. It writes $$\rho(s)\frac{\partial \rho(s^{-1})}{\partial z_{ij}} = \sum_{1 \leq \ell \leq n} s_{\ell i} \varepsilon_{\ell j} + s_{\ell j} \varepsilon_{\ell i}$$ for $$i \neq j$$, and $$\sum_{1 \leq \ell \leq n} s_{\ell i} \varepsilon_{\ell i}$$ for $$i=j$$. Here $$\varepsilon_{pq}$$ denotes the matrix with a $$1$$ in its $$pq$$ component and a $$0$$ in all other components, and represents the matrix it uses to act on $$V$$. When $$n=1$$, this reduces to $$sk$$ as we wanted since $$\varepsilon$$ acts as $$k$$.

I did an example for $$n=2$$ and the representation $$V \otimes V$$ of $$\operatorname{GL}_2$$, where $$V$$ is its standard representation. From this, I think the pattern may be $$\rho(s)\frac{\partial \rho(s^{-1})}{\partial z_{ij}} = \sum_{1 \leq \ell \leq n} s_{\ell i} \varepsilon_{j\ell} + s_{\ell i}\varepsilon_{j \ell}$$ for $$i = j$$, and $$\sum_{1 \leq \ell \leq n} s_{\ell i} \varepsilon_{i\ell}$$ for $$i=j$$. The fact that one of the indices is switched is not worrying, since $$s_{ij} = s_{ji}$$.

This possible answer feels like picking out a component (or two components) of the matrix $$\begin{pmatrix} \varepsilon_{11} & \dots & \varepsilon_{1n} \\ \vdots & \ddots & \vdots \\ \varepsilon_{n1} & \dots & \varepsilon_{nn} \end{pmatrix}\begin{pmatrix} s_{11} & \dots & s_{1n} \\ \vdots & \ddots & \vdots \\ s_{n1} & \dots & s_{nn} \end{pmatrix}$$ For $$i=j$$, we're picking out the $$ii=jj$$ component, and for $$i\neq j$$ we're summing the $$ij$$ component and the $$ji$$ component. I'm hoping to do another example (hopefully in the unitary case, so that $$s_{ij} \neq s_{ji}$$), but I don't think I know enough of the Lie group/Lie algebra theory to prove it even if the pattern continues.

EDIT again: I've now also done it for $$\operatorname{Sym}^3V$$, and it followed the same pattern. I did it (as much as I could) in the unitary case, using the representation $$V \otimes V$$ of the group $$\operatorname{GL}_2 \times \operatorname{GL}_2$$, where the first copy of $$\operatorname{GL}_2$$ acts on the first copy of its standard representation $$V$$, and the same for the second copies. Since I didn't know what to plug in for the derivatives of the inputs, I left them as $$a^\prime, b^\prime$$ and so on. In the end the terms paired up in a way that implies the pattern is present here as well, but it is difficult to see the actual final answer since I don't know how the coordinates vary with each $$z_{ij}$$. (In fact that was what I was hoping to get from this specific exercise, but the fact that I had to deal with the Lie algebra of $$\operatorname{GL}_2 \times \operatorname{GL}_2$$ gave enough of a wrinkle that I couldn't recover that from what I found.)