Let $G$ be a simple algebraic group over an algebraically closed field $k$. I believe all of the following invariants are well-defined. Besides the coxeter number, I haven't read about the others, so my notation may be nonstandard.
The rank of $G$, $\text{rk}(G)$, is the number of simple roots in the abstract root system associated to the simple lie algebra $\mathfrak{g}=\text{Lie}(G)$. The Coxeter number $h(G)$ is the total number of roots divided by $\text{rk}(G)$.
Define the nilpotency class of $G$ to be that of the unipotent radical $U$ of any Borel subgroup $B$. That is, let $\text{nc}(G)$ be the length of the lower central series of $U$.
Finally, let the nilpotency class of $\mathfrak{g}$ be the largest nilpotent index of nilpotent elements $x\in\mathfrak{g}$, ie, $\text{nc}(\mathfrak{g})$ is the smallest $m$ such that $(\text{ad}_x)^m\in\text{End}(\mathfrak{g})$ is $0$ for all nilpotent elements $x\in\mathfrak{g}$.
For $G=SL_n(k)$, it seems to me that $h(G)=\text{nc}(G)=\text{nc}(\mathfrak{g})=n$.
I guess I have two questions:
- Does the setup make sense? ie, is everything well-defined and are my computations for $SL_n(k)$ correct?
- For arbitrary simple $G$, does the equality $h(G)=\text{nc}(G)=\text{nc}(\mathfrak{g})$ hold?