multiplicity

《Class Field Theory》 · 第 242 页 · Section 9.1「Weights of Representations」章首约定及 Definition 9.1（省略中间内容）

Throughout the chapter, we assume that $\mathfrak g=\mathfrak k_{\mathbb C}$ is a complex semisimple Lie algebra and that $\mathfrak h=\mathfrak t_{\mathbb C}$ is a fixed Cartan subalgebra of $\mathfrak g$ (compare Proposition 7.11). We fix on $\mathfrak g$ an inner product that is real on $\mathfrak k$ and that is invariant under the adjoint action of $\mathfrak k$, as in Proposition 7.4. […] Definition 9.1. Let $(\pi,V)$ be a representation of $\mathfrak g$, possibly infinite dimensional. An element $\lambda$ of $\mathfrak h$ is a weight of $\pi$ if there exists a nonzero vector $v\in V$ such that $\pi(H)v=\langle\lambda,H\rangle v$ (9.1) for all $H\in\mathfrak h$. The weight space corresponding to $\lambda$ is the set of all $v\in V$ satisfying (9.1) and the multiplicity of $\lambda$ is the dimension of the corresponding weight space.