Let $K$ be a simply connected compact connected semisimple Lie group, and let $T\le K$ be a maximal torus. Let $S^1:=\{z\in\mathbb C:|z|=1\}$, let $X^*(T):=\operatorname{Hom}_{\mathrm{cts}}(T,S^1)$ be the character lattice of $T$, let $\mathfrak{k}$ be the [Lie algebra](/page/Lie%20Algebra) of $K$, let $\mathfrak{k}_{\mathbb C}:=\mathfrak{k}\otimes_{\mathbb R}\mathbb C$, let $R\subset X^*(T)$ be the root system of the complexified Lie algebra $\mathfrak{k}_{\mathbb C}$ with respect to $T$, and let $Q:=\mathbb ZR\subset X^*(T)$ be the root lattice. For $\alpha\in R$, let $\mathfrak{k}_{\mathbb C,\alpha}:=\{X\in\mathfrak{k}_{\mathbb C}:\operatorname{Ad}_t(X)=\alpha(t)X\text{ for every }t\in T\}$ be the corresponding root space. Use the standard simply connected identification $X^*(T)=P$, where $P$ is the weight lattice of $R$. Then evaluation defines a natural [group isomorphism](/page/Group%20Isomorphism) $Z(K)\cong \operatorname{Hom}(P/Q,S^1)$, namely the map sending $z\in Z(K)$ to the character $P/Q\to S^1$ given by $\lambda+Q\mapsto \lambda(z)$.

Fix a choice of positive roots $R^+\subset R$, and let $P_+\subset P$ be the corresponding dominant weight lattice. If $F\le Z(K)$ is a finite central subgroup, $G:=K/F$, and $\rho:K\to GL(V)$ is an irreducible finite-dimensional complex representation with highest weight $\lambda\in P_+$ and highest-weight line $L_\lambda\subset V$, then $\rho$ descends to a representation of $G$ if and only if every element of $F$ acts as the identity on $L_\lambda$.