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 define
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\begin{align*}
\mathfrak g_{\mathbb C}:=\mathfrak g\otimes_{\mathbb R}\mathbb C.
\end{align*}
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Define
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\begin{align*}
\mathfrak t_{\mathbb C}:=\mathfrak t\otimes_{\mathbb R}\mathbb C.
\end{align*}
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Let $\Phi\subset \mathfrak t_{\mathbb C}^*$ be the root system of $\mathfrak g_{\mathbb C}$ with respect to $\mathfrak t_{\mathbb C}$, and choose a positive root system $\Phi^+\subset \Phi$. For $\alpha\in\Phi$, let
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\begin{align*}
(\mathfrak g_{\mathbb C})_\alpha:=\{X\in\mathfrak g_{\mathbb C}:[H,X]=\alpha(H)X\text{ for every }H\in\mathfrak t_{\mathbb C}\}
\end{align*}
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denote the corresponding root space. Let
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\begin{align*}
\pi:G\to GL(V)
\end{align*}
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be a nonzero irreducible finite-dimensional complex representation, and let
denote the complex-linear extension of its derived representation. For $\mu\in\mathfrak t_{\mathbb C}^*$, define the weight space
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\begin{align*}
V_\mu:=\{v\in V:d\pi(H)v=\mu(H)v\text{ for every }H\in\mathfrak t_{\mathbb C}\}.
\end{align*}
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A weight of $V$ means an element $\mu\in\mathfrak t_{\mathbb C}^*$ such that $V_\mu\ne\{0\}$. Define the positive-root order on $\mathfrak t_{\mathbb C}^*$ by $\mu\preceq\nu$ if
for integers $n_\alpha\ge 0$, all but finitely many equal to $0$. Then there exists a unique weight $\lambda\in\mathfrak t_{\mathbb C}^*$ of $V$ such that $V_\lambda$ contains a nonzero vector $v$ satisfying
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\begin{align*}
d\pi(X)v=0
\end{align*}
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for every $\alpha\in\Phi^+$ and every $X\in(\mathfrak g_{\mathbb C})_\alpha$. This weight $\lambda$ is the unique maximal weight of $V$ for the order $\preceq$, equivalently the unique highest weight of $V$ with respect to $(T,\Phi^+)$, and