Weight space

Remarks

Example: let $\mathfrak{g} = \mathfrak{sl}_2(\C)$ act on $V = \C^2$, and let $\mathfrak{h}$ be spanned by $h = \operatorname{diag}(1, -1)$. For $\lambda(h) = 1$, the weight space $V_\lambda$ is the line $\C e_1$, since $h e_1 = e_1$ and the only eigenvectors of $h$ for $1$ are the multiples of $e_1$. Non-example: for $\lambda(h) = 2$, the space $V_\lambda$ is $0$, since $2$ is not an eigenvalue of $h$, so this $\lambda$ is not a weight. If $\mathfrak{h}$ is spanned by one element $h$, then $V_\lambda$ is the eigenspace of $\rho(h)$ for the eigenvalue $\lambda(h)$. The generalized weight space is the larger space of the $v$ such that for each $h \in \mathfrak{h}$ some power of $\rho(h) - \lambda(h)$ kills $v$. For nilpotent $\mathfrak{h}$ the space $V$ is the direct sum of these spaces when $F$ is algebraically closed of characteristic $0$ and $V$ is finite-dimensional.