I am recently learning about the Beilinson-Bernstein localization theorem and I am working on examples matching D-modules with Lie algebra representations. I would like to ask which D-module corresponds to the Verma module $M(0)$ of $\text{sl}_2$, whose highest weight is $0$.
Let $\mathbb{C}$ be our base field and let $G=\text{SL}_2$ and $X=G/B=\mathbb{P}^1$. Let $V=\mathbb{P}^1-\{[1:0]\}$ and $j:V\hookrightarrow X$ be the inclusion. I know that $\mathcal{N}(0):=j_*(\mathcal{O}_V)$ is the D-module corresponding to the dual Verma module $N(0)$, whose highest weight is $0$. Applying the duality functor $\mathbb{D}_X$, we get the D-module $\mathcal{M}(0):=\mathbb{D}_X j_*(\mathcal{O}_V)$ corresponding to the Verma module $M(0)$ of highest weight $0$.
However, the description above of $\mathcal{M}(0)$ seems abstract to me, particularly because I find it hard to compute the duality functor $\mathbb{D}_X$. Is there a more explicit description of $\mathcal{M}(0)$?
I am also curious about how we can compute $\mathbb{D}_X\mathcal{N}(0)$. I know the definition of duality functor, but I failed to compute it. Is there any reference that worked on this example?
Thank you
I found the answer.
D-module matching with M(0)