Exercise 11, § 11

Let $(x, h, y)$ be a principal $\mathfrak{s}\mathfrak{l}_2$-triplet in the semi-simple Lie algebra $\mathfrak{g}$. Give $\mathfrak{g}$ the $\mathfrak{s}\mathfrak{l}(2, k$)-module structure defined by this triplet. Show that the

module thus defined is isomorphic to $\bigoplus_{i=1}^lV(2k_i-2)$, where the $k_i$ are the

characteristic degrees of the algebra of invariant polynomial functions on $\mathfrak{g}$. (Reduce to the case in which $\mathfrak{g}$ is splittable simple. Use Cor. 1 of Th. 1 of §8, no. 3, and Chap. VI, §4, Exerc. $6c$)$.)^{20}$