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.


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).

  • $\begingroup$ @Chuck: The transition to prime characteristic is an intriguing possibility in this setting, though usually delicate. It's probably worth asking Bram Broer himself to explain his remark. And there is some further work beyond this paper which adapts some of Broer's arguments about normality to fields of good characteristic: Jesper Funch Thomsen, J. Algebra 227 (2000). $\endgroup$ – Jim Humphreys Mar 16 '12 at 17:55

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