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 $X^*(T):=\operatorname{Hom}_{\mathrm{cts}}(T,S^1)$ be the character group of $T$. 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 $R\subset X^*(T)$ be the root system of $\mathfrak g_{\mathbb C}$ with respect to $\mathfrak t_{\mathbb C}$, choose a positive root system $R^+\subset R$, and for each $\alpha\in R$ let $\alpha^\vee$ denote the corresponding coroot. Define the dominant integral weights for $(G,T,R^+)$ by $X^*(T)^+:=\{\lambda\in X^*(T):\langle \lambda,\alpha^\vee\rangle\ge 0\text{ for every }\alpha\in R^+\}$. For an irreducible continuous finite-dimensional complex representation $\rho:G\to GL(V)$, let $\lambda_V\in X^*(T)$ denote the maximal $T$-weight of $V$ with respect to the partial order generated by $R^+$, equivalently the weight of any non-zero vector annihilated by all positive root spaces in the differentiated $\mathfrak g_{\mathbb C}$-module. If $\operatorname{Irr}(G)$ denotes the set of isomorphism classes of irreducible continuous finite-dimensional complex representations of $G$, then the assignment $[V]\mapsto \lambda_V$ defines a bijection from $\operatorname{Irr}(G)$ onto $X^*(T)^+$.