Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$.
What is a set of representatives of the orbits of $\Delta$ under $f$ ?
Thanks!
Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$. What is a set of representatives of the orbits of $\Delta$ under $f$ ? Thanks! 

In a bunch of important cases, the orbits (of the group $\langle f \rangle$) acting on $\Delta$, correspond to roots in another root system. This is the procedure called "folding", which gives a way to reduce non simplylaced root systems to simplylaced ones. For example, $A_{2n1}$ folds to $C_n$, $D_{n+1}$ folds to $B_n$, $E_6$ folds to $F_4$. $D_4$ folds to $G_2$ (if you take an automorphim of order 3). If you start with $A_{2n}$, I guess the result is a nonreduced root system (i.e., one in which $\alpha$ and $2\alpha$ can both be roots for some $\alpha$), but I don't have a reference that includes nonreduced root systems handy, so I haven't confirmed that. 


$E_6$
the table at the end of Springer's 1966 paper in IHES No. 30, available online through www.numdam.org, is convenient). – Jim Humphreys Jan 10 '11 at 17:15