I´m working myself right now through this article and have a question regarding the proof of Proposition 6.4, case II) on page 345-346. I try to give a short outline of the situation:
We have $G \subset SL(V)$ a connected algebraic group, acting irreducible on a complex vectorspace $V$ and a reflection $r \in GL(V)$ which normalizes $G$ and has special eigenvalue $c \in \mathbb{C}^*$. In the proof we consider the Lie-algebra $\mathfrak{g}$ of $G$, which is semisimple and also acts irreducible on $V$. Now in said case II) we have, that $r-Id \in \mathfrak{g}$ is a nilpotent element. By application of the Jacobson-Morozov theorem, said nilpotent element is contained in a subalgebra $\mathfrak{s} \subset \mathfrak{g}$ with $\mathfrak{s} \simeq \mathfrak{sl}(2,\mathbb{C})$. Then the author states
Since the dimension of $ker(r-Id)=n-1$ we deduce by $\mathfrak{sl}(2)$-representation theory, that $\mathbb{C}^n \simeq \mathbb{C}^2 \oplus \mathbb{C}^{n-2}$ as an $\mathfrak{s}$ module.
And I was wondering, to what exactly he is refering to with $\mathfrak{sl}(2)$-representation theory.
Best Answer
Over the last few days you have asked a huge number of questions about this paper you are reading. I think you might gain more by spending a bit more time working on it yourself before asking for help.
In this case all they are saying is that if the kernel of $r-\mathrm{Id}$ is $(n-1)$-dimensional the only possible $n$-dimensional $\mathfrak{sl}_2$ rep with that property is the sum of a $(n-2)$-dimensional trivial rep $\mathbb{C}^{n-2}$ and the $2$-dimensional standard rep $\mathbb{C}^{2}$. Then $r-\mathrm{Id}$ is $0$ on $\mathbb{C}^{n-2}$ and has $1$-dimensional kernel on $\mathbb{C}^{2}$.
This is just from basic properties of $\mathfrak{sl}_2$ reps. The key observation is that any $\mathfrak{sl}_2$ irrep has a $1$-dimensional kernel (for any of its nilpotent elements in fact but perhaps more simply for $e$ and $f$ in the usual basis $e,f,h \in \mathfrak{sl}_2$).