Let $\sigma$ a finite-order automorphism of a finite-dimensional complex simple Lie algebra $\frak g$. Denote the order of sigma by $k$ and fix a $k^{th}$ root of unity.
It is well know that $k=1,2 \text{ or } 3$ and that $$\frak g=\oplus_{j\in \mathbb Z_k} \frak g_j$$ where $$\frak g_j=\left\{ x\in \frak g \mid \sigma(x)=w^jx\right\}.$$
Moreover, $\frak g_0$ is a simple Lie algebra.

QUESTION: Let $\lambda$ a weight of $\frak g$ and $V(\lambda)$ the irreducible representation of weight $\lambda$. Denote by $V(\lambda)_{\frak g_0}$ the $\frak g_0$-module obtained by restriction of the action of $\frak g$. Is $V(\lambda)_{\frak g_0}$ irreducible as a $\frak g_0$-module???

THANKS,

Note: The results mentioned can be found in the Kac book.