Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 第 107 页 · Chapter VI「Representation Theory」章首约定及 Section 20.1「Weight spaces」
Throughout this chapter $L$ will denote a semisimple Lie algebra (over the algebraically closed field $F$ of characteristic $0$), $H$ a fixed CSA of $L$, $\Phi$ the root system, $\Delta=\{\alpha_1,\ldots,\alpha_\ell\}$ a base of $\Phi$, $\mathcal W$ the Weyl group. […] If $V$ is a finite dimensional $L$-module, it follows from Theorem 6.4 that $H$ acts diagonally on $V$: $V=\coprod V_\lambda$, where $\lambda$ runs over $H^*$ and $V_\lambda=\{v\in V\mid h.v=\lambda(h)v\text{ for all }h\in H\}$. For arbitrary $V$, the subspaces $V_\lambda$ are still well-defined; whenever $V_\lambda\ne0$, we call it a weight space and we call $\lambda$ a weight of $V$ (more precisely, a “weight of $H$ on $V$”).