Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
As you say, "This result is pretty standard and can be found in many places." No need for Google here, since this site already has answers to the .question.
Two comments: 1) A search of this site for something like "ample line bundle" reveals many overlapping questions here, such as 99361. 2) It's probably better to use Chevalley's classification to pass to the algebraic setting over any algebraically closed field, since the answer to the question has little to do with the complex setting (or with combinatorics).
@Nicolas: This is a complicated subject, given the earlier distinction between strong and weak versions of BGG resolution. I'll give it some more thought, but meanwhile please check the list of revisions on my homepage or on the AMS bookstore page. (There is for example a modification in the proof method for Theorem 6.8 at the bottom of page 118.) Nowadays most publishers avoid corrected reprints, preferring the cheaper print-on-demand technology using the original plates.
The comolex seting adds nothing here. For sources, see the 1967-68 Steinberg lectures at Yale Lectures on Chevalley groups (in the somewhat edited and tpeset version now puboished by the AMS), or seeJantezn's book Representations of Algebraic Groups (now also published by the AMS), for a more comprehensive treatment.
It might help to point out that Steinberg's mimeographed lecture notes were distributed for many yeers by the Yae mathematics department, but a typeset and slightly revised set of notes was published after Steinberg's death by the AMS as part of their University Lecture Series
@Sebastian: "Abelian Lie group" is too broad here. Even Wallach's book Real Reductive Groups I gives an algebraic defini.tion of "real reductive Lie group". The problem is that a vector group is not a reductive Lie group (say over $\mathbb{C}$).
@Sebastian: I think the ambiguity comes into the proviso that the Lie algebra involves an abelian part. Does the additive Lie group qualify as "reductive", or the product of this group with a simple Lie group? This is pretty far from the Borel-Tits notion of reductive algebraic group.
