Representation of $\mathfrak{sl}_2(\mathbb{C})$ corresponding to Lie algebra representation

Question

We have a representation $R$ of a Lie group $\mathrm{SL}_2(\mathbb{C})$ in the space of polynomials $\mathbb{C}[x,y]$ such that $R\begin{pmatrix} a & b \\ c & d\end{pmatrix}(P(x,y))=P(ax+cy,bx+dy)$. How to describe the corresponding representation of the tangent Lie algebra $\mathfrak{sl}_2(\mathbb{C})$? In particular, how to describe the action of standard basis matrices $\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$,$\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$ and $\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}$?

Answer 1

Use the one-parameter subgroups $e^{At}$, $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, generated by these matrices via the exponential mapping and differentiate $R(e^{At})P$ at $t=0$.