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First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include …

It's worthwhile to explain something of the background, since Patrick Delorme's preprint never got published in full. It's a 23 page typed double-spaced document with symbols inserted by hand, dis …

It would help to recall Dynkin's definition of "regular semisimple subalgebra" of a complex semisimple Lie algebra (say $\mathfrak{g}$), which has not become standard in later literature. This just …

Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ r …

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 …

I'm not sure whether there is an efficient way to answer your question (or a written reference), but it's possible to analyze the situation case-by-case. It's probably best to start with an irreducib …

You are asking a number of related questions here, most of which require more reading of Lusztig's papers. See the reference list in my conference paper here, for example. But note first that the n …

Apparently this isn't discussed in any of the published literature, even in the numerous exercises for Bourbaki's Chapter VI on root systems in Lie Groups and Lie Algebras. I'm not sure how strong t …

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” reflectio …

I'll just add a few comments to what Jay has already said, in community-wiki style. 1) Most of these ideas have been developed over the past century, from E. Cartan onward, but at first in the Lie gr …

Rather than prolong the tangled comments, I'll try to provide a straightforward answer to the current formulation of the question. As noted already, there are two small cases of irreducible root sy …

It's important to emphasize that the simple group here is typically isomorphic to the rotation subgroup of $W$, which has index 2 and doesn't contain the reflections. So you need to look at the prec …

Maybe I can partially answer your second question by refocusing it somewhat. Carter's chapters 11-13 cover a lot of ground and were hard to organize in a straight line fashion, but the main theme is …

Rather than overload the question with side remarks, I'll add some background here in community-wiki format to indicate what can be gotten from Springer's relatively elementary treatment of regular el …

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl group …