Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 3

For $t\in \mathfrak{t}$, the root-space decomposition gives its centralizerEvery root space is one-dimensional, so this centralizer has the minimum possible dimension $\ell =\dim \mathfrak{t}$ exactly when no summand on the right occurs. By the Regular element criterion in a Cartan subalgebra,

Choose a positive system of a root system in which $\alpha$ is a simple root; this is possible after applying an element of the Weyl group. Let $\theta$ be the highest root. Since the rank is greater than one, $\theta \ne \alpha$, and the maximality of $\theta$ implies that $\theta +\alpha$ is not a root.

For $0\ne x\in {\mathfrak{g}}_{\alpha }$, the $(\ell -1)$-dimensional space $\ker \alpha \subset \mathfrak{t}$ centralizes $x$. The line $\mathbb{C}x$ also centralizes $x$, and $[x,{\mathfrak{g}}_{\theta }]\subseteq {\mathfrak{g}}_{\alpha +\theta }=0$. These independent spaces giveHence a nonzero simple-root vector is not regular when $\ell >1$.

Put $h_0=\psi (h)$ and $e_0=\psi (e)={\sum }_{i=1}^{\ell }{e}_i$, where $0\ne e_i\in {\mathfrak{g}}_{{\alpha }_i}$. Since $[h_0,e_0]=2e_0$ and the root spaces are a direct sum, ${\alpha }_i(h_0)=2$ for every simple root. Every root has simple-root coefficients of one sign, so no root vanishes on $h_0$. The Regular element criterion in a Cartan subalgebra therefore shows that $h_0$ is regular.

Restrict the Adjoint representation of $\mathfrak{g}$ along $\psi$. By Complete reducibility of semisimple Lie algebra representations, it is a direct sum of finite-dimensional sl2 Lie algebra modules. The $h_0$-eigenvalues on a root space are twice the heights of the roots, so they are all even. Each irreducible summand consequently has even highest weight, contains exactly one zero-weight vector, and has a one-dimensional kernel for the raising operator $e_0$ by the Classification of finite-dimensional sl2 representations.

Because $h_0$ is regular, its zero-weight space in $\mathfrak{g}$ is precisely $\mathfrak{t}$ and has dimension $\ell$. There are therefore exactly $\ell$ irreducible summands, whenceThus $e_0$ is regular; equivalently it is a principal nilpotent element in the given Principal sl2 subalgebra.