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To answer your question, it's enough to point out that every root (in a reduced root system such as $\Phi$) plays the role of simple root in some basis $\Delta$. Note however that some absolute root …

I'd be extremely surprised if such tables or database existed, mainly because the number of possible reduced decompositions for a Weyl group element tenda to grow very large as the rank increases. …

Let me add a few comments in community-wiki format. There doesn't seem to be a convenient reference, apart from the one in Digne-Michel which Jay Taylor cites. But even here, the authors don't give …

The three questions asked are fairly elementary, as the comment by LSpice indicates; in the format here, it's best to avoid multiple questions however. Aside from this, it's probably more natural t …

As Paul Levy's answer suggests, your question probably needs some case-by-case work to be answered completely. The most general perspective may come from older work of Borel-Tits in their 1972 IHES …

The answer to the question might be yes (over an algebraically closed field of any characteristic), though it's not clearly documented in the literature. First let me add a reference to the theorem …

Probably the recent paper by Herpel and Stewart here helps to settle your basic question positively. Though the correspondence between subgroups of Lie groups and Lie algebras in the classical situa …

Paul Garrett's book and lecture notes provide a reasonable approach to the subject, for which there are few textbook options. There are of course other lecture notes, usually slanted in some way whi …

Let me add a couple of things to what grghxy has said. 1) The study of these groups over $\mathbb{Z}$ has been complicated, going back to Chevalley's work and Borel's lecture notes (which aren't quit …

Leaving aside your minor changes in the notation and definition of Cabanes-Enguehard (who also adopt some arbitrary notation), my understanding of their exercise is that it responds to the obvious non …

I can't answer your question completely, but this extended comment may be helpful. Probably the answer will be "yes", judging at least from the old classification by Tits summarized in the proceedings …

One classical source for the case $n=2$, with somewhat old-fashioned notation for some of the related groups, is a paper by Richard Brauer and his student Cecil Nesbitt: On the modular characters of g …

EDIT: I didn't comment at first on your actual question, since I wasn't familiar enough with that passage in Borel-Tits. Their (3) strikes me as wrongly stated. Moreover, it doesn't seem to come up …

Since the comments are already getting long, I'll add this in community-wiki format to clarify a few points. I should emphasize that I'm not at all a specialist in GIT but have dealt with neighborh …

The edited question (and the answer to the first version) stilll leave me somewhat confused, so I'd suggest starting with concrete low-rank examples in order to focus better on the issues involved. ( …