Structure of projective indecomposable modules for $\mathfrak{sl}_2$

Question

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.

Answer 1

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 a 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 $w_{n}=y^{n}\cdot w$. The $v_{n}$-term in the action of $x$ on $w_{n}$ corresponds to "off-diagonal" matrix elements that store the information of a nontrivial extension. Then it's easy to check the action of the Casimir is not semisimple.

This calculation relies on the assumptions of category $\mathcal{O}$. There are other extensions with non-semisimple action of $h$, see the exercise in Page 48.