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

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

THANKS,

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