Weight of a representation

Remarks

The main case is a complex semisimple Lie algebra $\mathfrak{g}$ with a Cartan subalgebra $\mathfrak{h}$. Example: $\mathfrak{g} = \mathfrak{sl}_2(\C)$, $\mathfrak{h} = \C h$ with $h = \operatorname{diag}(1, -1)$, and $V = \C^2$. Here $h e_1 = e_1$ and $h e_2 = -e_2$, so the weights are the forms with $\lambda(h) = 1$ and $\lambda(h) = -1$, each of multiplicity one. Non-example: the form with $\lambda(h) = 0$ is not a weight of $\C^2$, since $h$ has no nonzero vector in its kernel. For the adjoint representation, the nonzero weights are the roots and $V_0 = \mathfrak{h}$. If $x \in \mathfrak{g}_\alpha$ is a root vector, then $x \cdot V_\lambda \subseteq V_{\lambda + \alpha}$. For a nilpotent $\mathfrak{h}$ one often uses generalised weight spaces, the vectors killed by a power of $h - \lambda(h)$ for every $h$, as in weight space.