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.



Structure of projective indecomposable modules for $\mathfrak{sl}_2$
Denoting the highest weight vectors of
$M(\lambda)$ and $M(\mu=-\lambda-2),\, \lambda\in\mathbb{Z}\_{>0}$ as $v$ and $v\_\mu$, we choose a preimage $w\in P(\mu)$ of $v\_\mu$ with $h\cdot w =\mu w$. Then for nontrivial extension the vector $x\cdot w$ does not vanish, hence $x\cdot w=A\_1 y^{\lambda}\cdot v$.


The action of $x$ can be calculated directly,


\begin{align}
& x\cdot v\_{n}=n(\lambda-n+1)v\_{n-1}\\
& x\cdot w\_{n}=A\_1 v\_{\lambda+n}-n (1 + \lambda + n)w\_{n-1}
\end{align}


where $v\_{n}=y^{n}\cdot v$ and $w\_{n}=y^{n}\cdot w$. Hence $x$ acts with Jordan blocks, and contributes nontrivially to the action of the Casimir.


This calculation relies on the assumptions of category $\mathcal{O}$. There are other extensions with non-semisimple action of $h$, which are more interesting in physics and are related to the discussion of logarithmic CFTs.


