Let $V$ be the adjoint representation space of $\mathfrak{sl}(2)$, over $\Bbb C$. Write the basis elements $\{e,f,h\}$ of $\mathfrak{sl}(2)$
Let $V_i=\{x\in V: [h,x]= ix\}$, then write: $$V=\bigoplus_{i\in \Bbb Z} V_i.$$
Why is it true that: $\text{ad} e:V_i\to V_{i+2}$ is injective for $i\leq -1$ and surjective for $i\geq -1$?
It is clear to me that $\text{ad} e$ does map between these spaces. As long as we are not looking at an element that is highest weight in every simple component, then $\text{ad} e$ will have trivial kernel. But if we are in $V_{-1}$ or lower, these are never highest weight elements, so this map must have trivial kernel. Is that true?
This is true, because it is true for every finite-dimensional irreducible $\mathfrak{sl}(2)$ representation and because every finite-dimensional $\mathfrak{sl}(2)$ representation is a direct sum of irreducible ones.