$$
H=\left(\begin{array}{cc}
1 & 0 \
0 & -1
\end{array}\right)
$$
in the Lie algebra sl $(2 ; \mathbb{C})$. Applying formula gives
$$
\left(\pi_{m}(H) f\right)(z)=-\frac{\partial f}{\partial z_{1}} z_{1}+\frac{\partial f}{\partial z_{2}} z_{2} .
$$
Thus, we see that
$$
\pi_{m}(H)=-z_{1} \frac{\partial}{\partial z_{1}}+z_{2} \frac{\partial}{\partial z_{2}} .
$$
Applying $\pi_{m}(H)$ to a basis element $z_{1}^{k} z_{2}^{m-k}$, we get
MATH0075 COURSE NOTES ：
We will use the following basis for $\operatorname{sl}(2 ; \mathbb{C})$ :
$$
H=\left(\begin{array}{cc}
1 & 0 \
0 & -1
\end{array}\right) ; X=\left(\begin{array}{ll}
0 & 1 \
0 & 0
\end{array}\right) ; Y=\left(\begin{array}{ll}
0 & 0 \
1 & 0
\end{array}\right)
$$
which have the commutation relations
$$
\begin{aligned}
&{[H, X]=2 X} \
&{[H, Y]=-2 Y} \
&{[X, Y]=H}
\end{aligned}
$$