Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 276 页 · Chapter V，Section 1「Weights」（省略中间内容）
Let $\mathfrak g$ be a complex semisimple Lie algebra. Fix a Cartan subalgebra $\mathfrak h$, and let $\Delta=\Delta(\mathfrak g,\mathfrak h)$ be the set of roots. […] Let $\varphi$ be a representation on the complex vector space $V$. Recall from §II.2 that if $\lambda$ is in $\mathfrak h^*$, we define $V_\lambda$ to be the subspace $\{v\in V\mid(\varphi(H)-\lambda(H)1)^n v=0\text{ for all }H\in\mathfrak h\text{ and some }n=n(H,V)\}$. […] The weight space corresponding to $\lambda$ is $\{v\in V\mid\varphi(H)v=\lambda(H)v\text{ for all }H\in\mathfrak h\}$, i.e., the subspace of $V_\lambda$ for which $n$ can be taken to be $1$. Members of the weight space are called weight vectors.