= Solution
Fix a <simple root> $\alpha_i$ and choose generators $e_i,f_i,h_i$ of the <sl2 subalgebra associated with a root>, with $h_i=\alpha_i^\vee$ and
$$
[h_i,e_i]=2e_i,\qquad[h_i,f_i]=-2f_i,\qquad[e_i,f_i]=h_i.
$$
The <highest-weight vector> satisfies $e_i v=0$ and $h_i v=m_i v$, where $m_i=\langle\lambda,\alpha_i^\vee\rangle$. The <classification of finite-dimensional sl2 representations> already implies that $m_i$ is a nonnegative integer. One can also see the necessary integrality directly from the <lowering operator>.
Indeed, $f_i^k v$, if nonzero, has $h_i$ eigenvalue $m_i-2k$. These distinct <eigenvalues> cannot occur indefinitely in a finite-dimensional space, so there is a largest $r\geq0$ with $f_i^r v\ne0$. The <sl2 Lie algebra> relations imply, by induction on $k$,
$$
e_i f_i^k v=k(m_i-k+1)f_i^{k-1}v.
$$
For $k=r+1$ the left side is zero, hence $(r+1)(m_i-r)f_i^r v=0$. Thus $m_i=r\in\mathbb Z_{\geq0}$. Since this holds for every <simple root>,
$$
\boxed{\lambda\text{ is a dominant integral weight}.}
$$
Back to article page
