# Global sections of the tangent bundle on Grassmannian and derivations

Let G=Gr(2,n) the grassmannian of 2-planes in $\mathbb{C}^n$, $R= \bigoplus_{k \geq 0} H^0(G, \mathcal{O}_G(k))$ its coordinate ring under the Plücker embedding, and consider the space of global section $H^0(G, T_G)$.

By Bott's theorem one has $H^0(G, T_G) \cong \mathfrak{sl}_n(\mathbb{C})$.

Now, I would like to get an explicit action of the generators of the algebra $\mathfrak{sl}_n(\mathbb{C})$ above on the coordinate ring $R$ as differential operators, but I did not manage to get something sensible. Is there any canonical way of doing this?

(Since I suspect this being something easy and well-known to Invariant theory people, I will put a 'reference request' tag as well)

• @JasonStarr yes, of course. Sorry for the typo, I will edit. – Enrico Jan 11 '17 at 23:27
• For every integer $k$, the derivation action of $H^0(G,T_G)=\mathfrak{sl}_n(\mathbb{C})$ on $H^0(G,\mathcal{O}_G(k))$ is equivalent to the irreducible $\mathfrak{sl}_n(\mathbb{C})$-module $\mathbb{S}^{(k,k)}((\mathbb{C}^n)^\vee)$, where $(k,k)$ is the partition of $2k$ into two parts of size $k$, and where $\mathbb{S}^{(k,k)}(-)$ is the corresponding Schur functor. (I am assuming that $n$ is at least $3$.) – Jason Starr Jan 11 '17 at 23:48