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

Notice removed Draw attention by Zhiyu

occurred Oct 19, 2019 at 18:37

Bounty Ended with Sam Gunningham's answer chosen by Zhiyu

occurred Oct 19, 2019 at 18:37

Tex edits

Source Link

edited Oct 18, 2019 at 20:22

LSpice

12.9k
4
45
69

Consider the natural action of $W_1=k\left<x,\frac{d}{dx}\right>$$W_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$, then. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentaillyessentially a $\mathfrak{sl_2}$$\mathfrak{sl}_2$-tuple ($[x\frac{d}{dx},\frac{d}{dx}] =-\frac{d}{dx} ,[x\frac{d}{dx},x^2\frac{d}{dx}]= x^2\frac{d}{dx}), [\frac{d}{dx},x^2\frac{d}{dx}]=2x\frac{d}{dx} )$$\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}$$\mathfrak{sl}_2$-module, and one easily checks $X$ is a non-split extension of $L(-1)$ by $L(1)$, for. 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 suchthis construction? Can we construct more examples of non-split extension of simple modules of simple lieLie algebras using differential operators on some good spaces such as symmetric spaces?

Consider the natural action of $W_1=k\left<x,\frac{d}{dx}\right>$ on $X=\mathbb C[x]$, then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentailly a $\mathfrak{sl_2}$-tuple ($[x\frac{d}{dx},\frac{d}{dx}] =-\frac{d}{dx} ,[x\frac{d}{dx},x^2\frac{d}{dx}]= x^2\frac{d}{dx}), [\frac{d}{dx},x^2\frac{d}{dx}]=2x\frac{d}{dx} )$.

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 such 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?

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?