Consider the natural action of $W_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentially a $\mathfrak{sl}_2$-tuple ($\left[x\frac{d}{dx},\frac{d}{dx}\right] =-\frac{d}{dx}$, $\left[x\frac{d}{dx},x^2\frac{d}{dx}\right]= x^2\frac{d}{dx}$, $\left[\frac{d}{dx},x^2\frac{d}{dx}\right]=2x\frac{d}{dx}$). 

So $X$ is a $\mathfrak{sl}_2$-module, and one easily checks $X$ is a non-split extension of $L(-1)$ by $L(1)$. For instance, $\mathbb C 1 \subseteq X=\mathbb C[x]$ is invariant and isomorphic to $L(1)$. Here $L(\lambda)$ is the the normalized simple module with highest weight $\lambda-1$.

How to generalize this construction? Can we construct more examples of non-split extension of simple modules of simple Lie algebras using differential operators on some good spaces such as symmetric spaces?