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Let $G$ be a semisimple connected complex Lie group, compact real Lie group or linear algebraic group. Let $\chi$ be the character of a finite dimensional irreducible representation of $G$ (I am particularly interested in the adjoint representation, or perhaps the fundamental ones, but general statements are useful too); lest there be any ambiguity, $\chi$ is considered a regular function on $G$ (viz., the trace of the representation).

I say that $g\in G$ is $\chi$-critical (for lack of a better term) when the differential of $\chi$ vanishes at $g$.

We can, of course, make this definition for $T\subset G$ a maximal torus, or for $\mathfrak{t}$ or $\mathfrak{g}$ the Lie algebras associated to $T$ and $G$. They all amount to essentially the same thing.

Basically, any information regarding $\chi$-critical (especially adjoint-critical) elements interests me, including a more standard term, or any mention of them in the literature. More specifically:

  • Is a $\chi$-critical element $g\in G$ that is semisimple (or equivalently, a $\chi$-critical $g\in T$) necessarily of finite order? If not, are there any natural conditions implying that it is? If yes, can we give a bound on their order or can we say something about it?

  • Is a $\chi$-critical element necessarily principal? "Principal" here is in the sense of belonging to a Kostant principal $\mathit{SL}_2$ or $\mathit{PGL}_2$: see e.g. Reeder, Torsion automorphisms of simple Lie algebras, §2.5, for definitions.

But again, other kinds of remarks concerning $\chi$-critical elements are welcome.

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  • $\begingroup$ Not that it narrows it down much, but I'm sure I've seen some variant of this somewhere in Steinberg … it's always a pleasant experience to look through his collected works, so no harm in doing so even if it's not there. :-) $\endgroup$ – LSpice Apr 26 '17 at 17:18
  • $\begingroup$ Presumably you mean "not principal" in your second question? For example, for a simple group every polynomial invariant on ${\mathfrak g}$ is of degree $>1$, so $0$ will be $\chi$-critical (focussing on the Lie algebra case, which I find easier to work with). $\endgroup$ – Paul Levy Apr 26 '17 at 17:52
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    $\begingroup$ Ok, let me reword that - $0$ is of course contained in any $\mathfrak{sl}_2$. The reason this jarred for me is that Kostant's criterion for regularity (read principality) of an element of the Lie algebra is that the differentials of a basic set of invariants are linearly independent. So vanishing of the differential would appear to me to be more associated with failure of principality. $\endgroup$ – Paul Levy Apr 26 '17 at 18:02
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    $\begingroup$ For relevant ideas in arbitrary characteristic, it's worth consulting the classic paper on regular elements by Steinberg (as indicated by L Spice): numdam.org/item/PMIHES_1965__25__49_0 Note especially how he generalizes Kostant's characteristic 0 work, which characterizes certain regular elements as those at which the differentials of all fundamental representations are independent. (In any case, you are asking too many open-ended questions here, which makes an "answer" elusive.) $\endgroup$ – Jim Humphreys Apr 26 '17 at 23:50
  • $\begingroup$ @JimHumphreys is right; that's what I was remembering. $\endgroup$ – LSpice Apr 28 '17 at 1:12
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The question "does every semisimple $\chi$-critical $g\in G$ necessarily have finite order?" (with $\chi$ being the character of the adjoint representation) has a negative answer, as shown by the following fairly trivial counterexample:

  • Let $g \in \mathit{SO}_6$ be the rotation that is the orthogonal direct sum (i.e., the block diagonal matrix) of the following three plane rotations: one of angle $0$ (i.e., the $2\times 2$ identity matrix), one of angle $\pi$, and one of angle $\theta$ arbitrary. The adjoint character¹ of $g$ is $-1$ (the smallest possible), it is critical, but $\theta$ can be arbitrary, so $g$ can fail to be of finite order.

Now this $\chi$-critical point is not isolated, so one might hope to salvage the question by asking "does every semisimple isolated $\chi$-critical $g\in T$ necessarily have finite order?" or parhaps "is every $\chi$-critical value necessarily witnessed by a $g\in G$ of finite order?". But the answer is still negative, as shown by the slightly following more interesting example of a $\chi$-critical rotation:

  • Let $g \in \mathit{SO}_{10}$ be the rotation that is the orthogonal direct sum of the following five plane rotations: one has angle $0$ and four have the same angle $\theta$ such that $\cos\theta = -\frac{1}{3}$. Then $\chi(g) = \frac{7}{3}$, it is isolated critical (in a maximal torus $T$ containing it), yet $g$ is not of finite order; furthermore, all $\chi$-critical elements with this particular critical value are similar to the $g$ I described, so none is of finite order.

This all results from not-particularly-enlightening computations. I have very little intuition on what makes a $g$ such as above remarkable, or, in the end, what "$\chi$-critical" really tells us.

The same counterexamples can be used against my question about principality.

One question I still have is whether $\chi$-critical values are necessarily rational, but I think I had better think more about it before I open a new question.

  1. Recall that if $g\in\mathit{SO}_{2n}$ is orthogonal direct sum of rotations of angles $\theta_1,\ldots,\theta_n$, then its adjoint character $\chi(g)$ is: $n+2\sum\cos(\theta_i\pm\theta_j)$ where the sum is on all $n(n-1)$ combinations of an unordered pair $\{i,j\}$ with $i\neq j$ together with a sign $\pm$.
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    $\begingroup$ Why don't you rather edit the question? $\endgroup$ – YCor May 5 '17 at 10:56
  • $\begingroup$ @YCor Because now I'm not sure what question I want to ask, except for a very vague "what can we say about $\chi$-critical elements or values?", which as Jim Humphreys pointed out, is really too broad; and if I figure out something more interesting, it might be quite different. In the mean time, I thought I might as well answer the question I did ask, because I learned something in the process (and since someone marked this question as favorite, I guess they want to hear the answer). $\endgroup$ – Gro-Tsen May 5 '17 at 12:33

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