Construction of non-split extension of simple modules of Lie algebras using linear differential operators

Question

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?

Answer 1

Perhaps you are already aware of this, but your observation is essentially an example of the Beilinson–Bernstein theory connecting $\mathfrak g$-modules to D-modules on the flag variety.

In your example, the 3 first-order operators you have written down extend over the point at $\infty$ to define global (holomorphic) vector fields on $\mathbb{CP}^1$, and in fact they span the space of such vector fields. Moreover, the algebra of global differential operators on $\mathbb{CP}^1$ is isomorphic to the central quotient $U_0(\mathfrak{sl}_2)$ (the quotient of the enveloping algebra by the maximal ideal of the center given by the annihilator of the trivial module), extending your observation that the three vector fields you wrote down satisfy an $\mathfrak{sl}_2$ relation.

More generally, the algebra of global differential operators on a flag variety $G/B$ (for a reductive algebraic group $G$ over $\mathbb C$ say) turns out to be isomorphic to the central quotient $U_0(\mathfrak{g})$. The localization theorem says that there is an equivalence of categories between modules for this algebra and sheaves of D-modules on $G/B$. Moreover, there is a nice theory which explains how many of your favourite $\mathfrak{g}$-modules look as $D$-modules.

In your case, the module $\mathbb{C}[x]$ can be thought of as the $D$-module $j_\ast \mathcal O$ on $\mathbb{CP}^1$, where $j$ is the inclusion of the open subset $\mathbb{A}^1 \hookrightarrow \mathbb{P}^1$. The general theory says that this should correspond to the dual Verma module (I think!) with highest weight $0$ (which is an extension of two simples as you stated). More generally, dual Verma modules can be realized as $\ast$-pushforwards of the structure sheaf on Bruhat cells in $G/B$, Verma modules as $!$-pushforwards, and simple modules as intersection cohomology extensions.

If you want to read more about this, I think the book by Hotta, Takeuchi, and Tanisaki called "D-Modules, Perverse Sheaves, and Representation Theory" (MSN) is a good place to look.