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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 …

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 …

Thanks for adding some further context. Maybe I can partly answer your question by making a series of comments: 1) At the beginning it's important to specify that the Borel subgroup $B_0$ is $F$-st …

There are several questions being asked (and an unexplained reference to a field of definition), but the answer to at least one of them is no: Take $G_1 = G_2 = \mathrm{SL}_3(\mathbb{C})$, with a give …

Angelo has indicated the most classical type of counterexample involving two semisimple elements of $\mathrm{PGL}_n$, but the question warrants some further discussion. Though it is posed (and maybe …

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 …

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. ( …

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 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 …

The answer to your question is yes. Your element $d$ is regular semisimple since its centralizer is assumed to have minimal dimension equal to the rank (being the maximal torus $T$). In turn, it's …

[REVISION]: After looking closer at the literature related to Deligne-Lusztig's Annals of Math. 103 (1976) paper, I'm more persuaded that the answer to your question is yes. In their paper, see in …

To supplement the answer given by Francois, I'd emphasize that the question is essentially algebraic (over an algebraically closed field of any characteristic) in the spirit of the Borel-Chevalley str …

To provide a more balanced context for the question (and the answer by pm), it's useful to separate the elementary notion of BN-pair from the far more sophisticated structure theory of algebraic group …

Here you are working over $\mathbb{C}$ (or perhaps any other splitting field of characteristic 0 for $G$). So the representation you are starting with is just the direct sum over all characters $\ch …

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 …