I'm reading BGG Category $\mathcal{O}$ by Humphreys.

In section 3.12 we look into the projective over $\mathfrak{sl}(2,\mathbb{C})$. If $\lambda\in\mathbb{Z}$ is a weight and $\mu=-\lambda-2$ is the other weight linked to $\lambda$ then from calculation and the BGG reciprocity we get that the projective cover $P(\mu)$ of $L(\mu)$ is in the non-split extension $$0\to M(\lambda)\to P(\mu)\to M(\mu)\to0$$ where $M(\lambda),M(\mu)$ are the Verma's modules.

After that, there is an exercise to calculate the action of the Casimir element $z:=h^2+2h+4yx$ on $P(\mu)$. I know how to prove that $(z-c)^2=0$ when $c:=\chi_\lambda(z)=\lambda^2+2\lambda$. I also know how to prove that $z$ doesn't act as a scalar, but I don't know how exactly it acts.

Also, I would like to know more details about the structure of $P(\mu)$, and how $x,y$ acts on the nontrivial/interesting weight spaces (coming from $M(\mu)$). Is there a general way to describe this? I try to compute but it seems not a lot of fun :(

Thank you!

I'm reading "BGG category $\mathcal{O}$" by Humphreys.

In section 3.12 we look into the projective modules over $\mathfrak{sl}(2,\mathbb{C})$. If $\lambda\in\mathbb{Z}$ is a weight and $\mu=-\lambda-2$ is the other weight linked to $\lambda$ then from calculation and the BGG reciprocity we get that the projective cover $P(\mu)$ of $L(\mu)$ is in the non-split extension $$0\to M(\lambda)\to P(\mu)\to M(\mu)\to0$$ where $M(\lambda),M(\mu)$ are the Verma's modules.

After that, there is an exercise to calculate the action of the Casimir element $z:=h^2+2h+4yx$ on $P(\mu)$. I know how to prove that $(z-c)^2=0$ when $c:=\chi_\lambda(z)=\lambda^2+2\lambda$. I also know how to prove that $z$ doesn't act as a scalar, but I don't know how exactly it acts.

Also, I would like to know more details about the structure of $P(\mu)$, and how $x,y$ acts on the nontrivial/interesting weight spaces (coming from $M(\mu)$). Is there a general way to describe this? I try to compute but it seems not a lot of fun.