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.