**Background**

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0. Let $B \subseteq G$ be a Borel subgroup and let $U \subseteq B$ be its unipotent radical. Let $\mathfrak g$, $\mathfrak b$, and $\mathfrak n$ be the corresponding Lie algebras. Also let $\mathcal T^*$ denote the cotangent bundle of the flag variety $G/B$; then $\mathcal T^*$ is isomorphic to the homogeneous bundle $G \times^B \mathfrak n$.

For any representation $M$ of $B$ let $\mathcal L(M)$ denote the homogeneous line bundle on $G/B$ with fiber $M$ and let $\mathcal L'(M)$ denote the pullback of this bundle to $\mathcal T^*$. In particular, for any weight $\lambda$ of $G$, we have the bundles $\mathcal L(\lambda)$ and $\mathcal L'(\lambda)$. Let $H^0(\lambda)$ denote the global sections of $\mathcal L(\lambda)$.

By the projection formula, we have $$ H^0( \mathcal T^*, \mathcal L'(M) ) \cong H^0 \big( G/B, \mathcal L( k[\mathfrak n] ) \otimes \mathcal L(M) \big) . $$ In particular, since $k[\mathfrak n]$ is a graded $B$-module, $H^0( \mathcal T^*, \mathcal L'(M) )$ naturally has the structure of a graded $G$-module.

**Question**

Let $\lambda$ be a weight of $G$. The natural $B$-equivariant surjection $k[\mathfrak g] \twoheadrightarrow k[\mathfrak n]$ corresponding to the inclusion $\mathfrak n \hookrightarrow \mathfrak g$ induces a sheaf surjection $\mathcal L(k[\mathfrak g]) \twoheadrightarrow \mathcal L(k[\mathfrak n])$ and hence, twisting by $\lambda$ and taking global sections, we obtain a $G$-equivariant gradation-preserving morphism $$ k[\mathfrak g] \otimes H^0(\lambda) \cong H^0( G/B, \mathcal L(k[\mathfrak g]) \otimes \mathcal L(\lambda) ) \to H^0( G/B, \mathcal L(k[\mathfrak n]) \otimes \mathcal L(\lambda) ) \cong H^0( \mathcal T^*, \mathcal L'(\lambda) ) . $$ In his paper "Line Bundles On The Cotangent Bundle Of The Flag Variety," Broer proves that this morphism is surjective when $\lambda$ is dominant. He also remarks that there is an alternate proof of this fact using the following standard fact from algebraic geometry: If $i : X \to \mathbb P^n_A$ is a morphism of $A$-schemes, then $i^*( \mathcal O(1) )$ is an invertible sheaf on $X$ which is globally generated by the pullback of the global sections of $\mathcal O(1)$.

My question is: what is this alternate proof? It's mysterious to me where this remark comes from. I assume that one should use the inclusion $$ G \times^B \mathfrak n \hookrightarrow G \times^B \mathfrak g \cong G/B \times \mathfrak g, $$ but I can't seem to figure the rest out. (My motivation is that I'd like to give a proof of this fact when $k$ is a field of arbitrary characteristic).