Highest weight

Remarks

Since $\le$ is a partial order, a highest weight is unique when it exists. A highest weight vector $v$ satisfies $x \cdot v = 0$ for every root vector $x \in \mathfrak{g}_\alpha$ with $\alpha \in \Phi^+$, since $\lambda + \alpha$ is not a weight. Example: for $\mathfrak{sl}_n(\C)$ with diagonal $\mathfrak{h}$, the module $\C^n$ has weights $\varepsilon_1, \dots, \varepsilon_n$, where $\varepsilon_i$ reads the $i$-th diagonal entry. With simple roots $\alpha_i = \varepsilon_i - \varepsilon_{i+1}$, one has $\varepsilon_1 - \varepsilon_j = \alpha_1 + \dots + \alpha_{j-1}$, so $\varepsilon_1$ is the highest weight and $e_1$ is a highest weight vector. Non-example: for $\mathfrak{sl}_2(\C)$, identify weights with numbers by $\lambda \mapsto \lambda(h)$, so that the simple root is $2$. The module $\C^2 \oplus \mathfrak{sl}_2(\C)$ has weights $1, -1$ and $2, 0, -2$; since $2 - 1 = 1$ is not an even integer, $1 \not\le 2$, and there is no highest weight. Each finite-dimensional irreducible module has a highest weight, and so does each module generated by a weight vector killed by all positive root vectors, such as a Verma module.