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Let $V$ be a complex vector space. Consider the algebra $D(\mathbb{P}(V),\mathcal{O}(n)))$ of global differential operators from line bundle $\mathcal{O}(n)$ to itself, here $n \in \mathbb{Z}_{\geqslant 0}$.

Can we define this algebra via generators and relations?

It looks like we can indeed describe the kernel of the natural map $U(\mathfrak{gl}(V)) \rightarrow D(\mathbb{P}(V), \mathcal{O}(n))$. This is a quantisation of the set of relations defining the closure of the minimal nilpotent orbit in $\mathfrak{sl}(V) \subset \mathfrak{gl}(V)$. Using now that the kernel of $U(\mathfrak{gl}(V)) \rightarrow D(\mathbb{P}(V), \mathcal{O}(n))$ should be zero being restricted to the affine chart in $\mathbb{P}(V)$ (corresponds to the fact that this kernel is an annihilator of a certain parabolic Verma) we can compute and get \begin{multline*} D((\mathbb{P}(\mathbb{C}^{m+1}), \mathcal{O}(n)))=\\ =U(\mathfrak{gl}_{m+1})/(e_{ki}e_{lj}-e_{li}e_{kj}-\delta_{i,l}e_{kj}+\delta_{i,k}e_{lj},\, (\sum_{p=1}^{m+1}e_{pp})-n ,\, i<j ,\, k<l), \end{multline*} here $e_{ij},\, 1 \leqslant i,j \leqslant m+1$ is the standard basis in $\mathfrak{gl}_{m+1}$.

The relation $\sum_{p=1}^{m+1}e_{pp}=n$ is clear and the relations $e_{ki}e_{lj}-e_{li}e_{kj}-\delta_{i,l}e_{kj}+\delta_{i,k}e_{lj}$ can be easily seen if one considers the natural action $\mathfrak{gl}(V) \curvearrowright V^* \setminus \{0\}$ and looks at the kernel of the natural map $U(\mathfrak{gl}(V)) \rightarrow D(V^* \setminus \{0\})$, now $D(\mathbb{P}(\mathbb{C}^{m+1}),\mathcal{O}(n))$ is just the hamiltonian reduction of $D(V^* \setminus \{0\})$ by the group $\mathbb{C}^\times$.

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  • $\begingroup$ If I am not mistaken, for $n=0$, this is $U(\mathfrak{sl}(V))/ (\ker \chi_0)$, where $\chi_0 : Z \to \Bbb C$ is the trivial character and $Z$ the center. But I am not sure how to write down the center in term of the usual Chevalley generators. $\endgroup$ Oct 25, 2020 at 22:40
  • $\begingroup$ I would say that you get $U(\mathfrak{sl}(V))/\operatorname{ker}\chi_0$ if you consider global differential operators on the flag variety of $\mathfrak{sl}(V)$. But I am asking about differential operators on $\mathbb{P}(V)$ that is a certain partial flag variety of $\mathfrak{sl}(V)$. $\endgroup$
    – Vas
    Oct 25, 2020 at 23:03
  • $\begingroup$ Yes, sorry I was too hasty. $\endgroup$ Oct 26, 2020 at 8:26
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    $\begingroup$ It is still true that there is a surjective algebra map $U(\mathfrak{sl}(V))\to D(\mathbb{P}(V),\mathcal{O}(n))$ for essentially the same reason as for the full flag variety and the case $n=0$. However I don't know if there is a good description of generators of the kernel. $\endgroup$ Oct 26, 2020 at 11:27

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