Hello,
I would like to get references or answers, for the following. How do I show that the anti-canonical line bundle (i.e. dual to top wedge power of cotangent bundle) on a flag variety (of a semisimple algebraic group in char. 0) is ample?
Thank you, Sasha
It's worth pointing out that the criterion for ampleness applies uniformly for reductive groups and flag varieties in any characteristic. From the algebraic group viewpoint, the only subtle point is that the flag variety $G/B$ should involve a Borel subgroup $B$ corresponding to negative rather than positive roots. Here you have a fixed maximal torus $T\subset B$ whose character group $X(T)$ contains the root lattice.
Classical ideas in algebraic geometry related to this set-up in characteristic 0 are used by Demazure in his 1976 Invent. Math. paper "A very short proof of Bott's theorem". Choosing $B$ in this nonstandard way avoids a sign change in the weight which determines a line bundle. With this convention (opposite to yours), ample line bundles correspond precisely to dominant weights which are also regular (so that some large enough positive multiple is very ample). In particular, you are concerned about the weight $2\rho$: here $2\rho$ is the sum of all positive roots, corresponding to the highest wedge product of the positive part of the Lie algebra. It's an elementary observation about roots and weights that $\rho$ itself is both dominant and regular (the smallest such weight in the usual partial ordering). So it already defines an ample line bundle whenever $\rho \in X(T)$, e.g., when the derived group of the reductive group is simply connected.
All of this machinery carries over intact to prime characteristic, where it's exploited by Kempf and then by Henning Andersen to explore the cohomology of line bundles on a flag variety. One textbook reference worth consulting is Section II.4.4 of Jantzen's large AMS book (second edition, 2003): Representations of Algebraic Groups. There are many earlier references I could point out in the work I've mentioned. In any case, the "right" answer to the question asked depends partly on what language is most comfortable in this interface of algebraic geometry and algebraic groups. But the question itself is fairly narrow and doesn't require the detailed study of Schubert varieties.
ADDED: Here are a few (incomplete) remarks about history and sources. Papers by Demazure in the 1960s shifted the study of cohomology of line bundles on flag varieties into the realm of algebraic geometry, though he stayed mostly in characteristic 0. Andersen's many papers from the 1970s onward exploit creatively the ampleness of a line bundle attached to a regular dominant weight in prime characteristic. He was inspired by Demazure's proof of Bott's theorem and also by Iversen's foundational paper in Advances in Mathematics (1976), which showed how to formulate many ideas about algebraic geometry and algebraic groups over an arbitrary algebraically closed field. Meanwhile Kempf in the 1970s proved some deep results applicable to prime characteristic in this framework. His long 1978 Advances paper "The Grothendieck-Cousin complex of an induced representation" treats line bundles in detail in part I (see especially Lemma 5.3). For me his papers are quite challenging to read, however. The reference I gave to Jantzen's Chapter II.4 (on Kempf's vanishing theorem) deals more concisely with ampleness of line bundles in 4.3 and 4.4. (Here the approaches of Andersen and Haboush to Kempf's theorem are explained.)
CODA: I should return to the short question originally asked. The essential point is to identify the weight $\chi \in X(T)$ which determines the line bundle in question on a flag variety $G/B$ where $T$ is a maximal torus in $B$ and $B$ corresponds (say) to negative roots of $G$ relative to $T$. This requires extracting $\chi$ from the formalism of the cotangent bundle: this bundle involves the subspace $\mathfrak{n}$ of the Lie algebra (= tangent space) which is the sum of positive root spaces; here one knows that $\dim \mathfrak{n} = N = \dim G/B$, so the $N$th exterior power carries the (regular dominant) weight $2\rho$ = sum of positive roots. Note too that duality doesn't affect this particular weight.
Let me try to add to Jim Humphreys's comprehensive answer by pointing out that the (very) ampleness of line bundles coming from regular dominant weights was already observed by Borel and Weil, and essentially appears as Thm.4 in Serre's Bourbaki report
Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d'après Armand Borel et André Weil). Séminaire Bourbaki, Vol. 2, Exp. No. 100, 447–454, Soc. Math. France, Paris, 1995.
(See the example following Thm.4 on p.453.)
Although Borel and Weil work over $\mathbb C$ their argument is probably worth summarizing here. They begin by observing that the line bundle $L_\lambda \to G/B$ coming from a weight $\lambda$ is spanned by global sections iff $\lambda$ is dominant. In this case the image of the map $G/B \to \mathbb P (H^0(G/B,L_\lambda)^\ast)$ defined by sections is of the form $G/P$ where $P\supset B$ is the parabolic subgroup defined by the simple roots perpendicular to $\lambda$. Thus if $L_\lambda$ is not very ample, so that the aforementioned fibration $G/B \to G/P$ is not an embedding, one concludes that $P$ must properly contain $B$ and consequently that $\lambda$ must be perpendicular to some simple root, hence is not regular.
