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 reducible as a $g_0$-module???g_0$-module for all$\lambda$? THANKS, Note: The results mentioned can be found in the Kac book. 4 added 11 characters in body; deleted 12 characters in body Let$\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$g_j={ x \in g \mid \sigma(x)=\omega^jx }.$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.

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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 restriction of restricting the action of $g$. 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.

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