Does every nilpotent lie in the span of simple root vectors

Question

Let $G$ be a reductive group (for simplicity, we can work over $\mathbb{C}$). Let $N \in \operatorname{Lie} G$ be nilpotent. Does there exist a Borel pair $(B,T)$ of $G$ such that $N$ lies in the span of the simple root vectors for $(B,T)$? This is true for $\mathrm{GL}_n$ by considering the Jordan form of a nilpotent matrix.

Answer 1

Let $\mathfrak g$ have Cartan subalgebra $\mathfrak h$, let $X_\alpha\in\mathfrak g_\alpha$ be a non-zero vector. Then you are looking at nilpotent elements of the form $\sum_{\alpha\in \Theta}X_\alpha$ for some subset $\Theta\subset\Delta$.

Note that each subset $\Theta\subset \Delta$ we can associate a parabolic subalgebra $\mathfrak p_\Theta\subset \mathfrak g$ generated by $\mathfrak h$ and $\mathfrak g_\alpha$ where $\alpha\in\Delta\cup(-\Theta)$, with Levi subalgebra $\mathfrak l_\Theta$ generated by $\mathfrak h$ and $\mathfrak g_\alpha$ for $\alpha\in\Theta\cup(-\Theta)$. Now from e.g. Section 4.1 of Collingwood-McGovern we see that the orbit of $\sum_{\alpha\in\Theta}X_\alpha$ is the principal nilpotent orbit of $\mathfrak l_\Theta$.

Thus, the nilpotent orbits you obtain are precisely those whose intersection with some Levi subalgebra of $\mathfrak g$ is principal.

Many nilpotent orbits do not satisfy this. For example, in type $C_n$ nilpotent orbits are parametrized by partitions of $2n$ with odd parts occuring with even multiplicity. However, the Levi subalgebras of $C_n$ are of the form $A_{i_1}\times\cdots \times A_{i_m}\times C_k$ so the nilpotent orbits obtained from them correspond to partitions where all but one part appears with even multiplicity (so the partition $[4,2]$ in type $C_3$ does not arise from your construction).