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](https://www2.bc.edu/mark-reeder/Torsion.pdf), §2.5, for definitions.


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



Critical points of characters on semisimple groups
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 $\pm$.


