Let $G$ be a compact connected Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $T\le G$ be a maximal torus with Lie algebra $\mathfrak t$, and let $\mathfrak g_{\mathbb C}:=\mathfrak g\otimes_{\mathbb R}\mathbb C$ and $\mathfrak t_{\mathbb C}:=\mathfrak t\otimes_{\mathbb R}\mathbb C$. Let $U(1):=\{z\in\mathbb C:|z|=1\}$, and let $X^*(T):=\operatorname{Hom}_{\mathrm{cts}}(T,U(1))$ denote the group of continuous torus characters. For each $\alpha\in\mathfrak t_{\mathbb C}^*$, define

\begin{align*} \mathfrak g_\alpha:=\{X\in\mathfrak g_{\mathbb C}:[H,X]=\alpha(H)X\text{ for every }H\in\mathfrak t_{\mathbb C}\}. \end{align*}

Let $R\subset \mathfrak t_{\mathbb C}^*$ be the nonzero root system of $\mathfrak g_{\mathbb C}$ with respect to $\mathfrak t_{\mathbb C}$, meaning the finite set of nonzero $\alpha$ such that $\mathfrak g_\alpha\ne\{0\}$, and choose a positive root system $R^+\subset R$. Let $V$ be a nonzero finite-dimensional complex [vector space](/page/Vector%20Space), and let $\rho:G\to GL(V)$ be a smooth finite-dimensional complex representation. Let $d\rho_{\mathbb C}:\mathfrak g_{\mathbb C}\to \mathfrak{gl}(V)$ be the complex-linear extension of the differential of $\rho$ at the identity. For each $\nu\in \mathfrak t_{\mathbb C}^*$, define the $\nu$-weight space by

\begin{align*} V_\nu:=\{v\in V:d\rho_{\mathbb C}(H)v=\nu(H)v\text{ for every }H\in\mathfrak t_{\mathbb C}\}. \end{align*}

Then there exist a weight $\lambda\in\mathfrak t_{\mathbb C}^*$ with $V_\lambda\ne\{0\}$ and a vector $v\in V_\lambda\setminus\{0\}$ such that

\begin{align*} d\rho_{\mathbb C}(X)v=0 \end{align*}

for every $\alpha\in R^+$ and every $X\in\mathfrak g_\alpha$. Equivalently, $v$ is a highest weight vector for $V$ relative to $(T,R^+)$.