User angelo - MathOverflow

representation theory and finite order automorphisms

2011-03-30T05:09:57Z

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 $V(\lambda)_{g_0}$ reducible as a $g_0$-module for all $\lambda$?

THANKS,

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

Comment by Angelo

2011-04-03T04:06:45Z

@Jim: I wrote something wrong in my last comment, sorry you and Bruce!. I was talking only about irreducible highest weight modules and I used the term UNIVERSAL. The term universal is valid and important in another context that was on my mind, about representations of loop algebras. On the other hand, I know these modules are infinite dimensional and rarely irreducible. But I disagree that the question is out of context.

Comment by Angelo

2011-03-30T13:59:03Z

@Mariano I am sorry, I didn't understand your question. Can you redo it? Thanks,

Comment by Angelo

2011-03-30T13:57:01Z

@Bruce: Ok, I agree that it may be the answer. But are you saying the answer is negative for all $g$ of type $A_n$, $D_n$ and $E_6$? (all types that admit a non trivial automorphism) Please, clarify me why! I guess that a bit of calculation is necessary, is not it? For instance, if $g=sl_3$ what we have ?