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Let $\sigma$ be a nontrivial Dynkin diagram automorphism of a finite-dimensional complex simple Lie algebra $\frak g$ (of type A, D or E) and let $\frak h$ be a Cartan subalgebra of $\frak g$. Let $I$ be a index set for the simple roots of $\frak g$ and $R$ the set of roots of $\frak g$. Consider the automorphism of $\frak g$ induced by $\sigma$ given by $\sigma(x_i^\pm)=x_{\sigma(i)}^\pm$ for all $i\in I$. It is well known that the order of $\sigma$ is 2 or 3 and we shall denote it by $m$. Fix a primitive $m^{th}$ root of unity $\xi$.

Consider ${\frak g}_j = \{ x\in {\frak g} \mid \sigma(x)=\xi^j x\}$. It means that $\frak g_0$ is the set of fixed points of this automorphism. It is also well known that each $\frak g_j$ is a $\frak g_0$-module and we denote its set of weights as a $\frak g_0$-module by $wt(\frak g_j)$. Fix $\frak h_0 = \frak g_0 \cap \frak h$.

How to prove the following:

1) If $\mu \in wt(\frak g_j)$ is non zero, then $\mu = \alpha|_{\frak h_0}$ for some $\alpha \in R$.

2) Let $\alpha,\beta\in R$. Then, $\alpha|_{\frak h_0} = \beta|_{\frak h_0}$ if and only if $\alpha=\sigma^j(\beta)$ for some $j$.

What could be a good reference for this kind of question?

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I would approach this kind of a question by building up some intuition working out a case, say type $A_2$, by hand. I think that would be faster than looking for a reference. But I have never done this exercise, so I may be wrong. –  Jyrki Lahtonen Mar 6 '12 at 15:47
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1 Answer 1

Most of the details of the setup you describe can be found in Chapter 8 of Kac's book Infinite dimensional Lie algebras. The motivation there is that this is important in the study of twisted loop (and affine) Lie algebras.

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