Highest weight vector

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Let $V$ be a representation of a Lie algebra $\mathfrak{g}$ , resp. a Lie group $G$ . A weight vector $v_\lambda \in V$ of weight $λ$ is called a highest weight vector, or vector of highest weight, if all other weights of $V$ are smaller (see ordering of weights) than $λ$ .

Usually, it is assumed that a highest weight vector is a highest weight of a highest weight module (i.e. the representation is generated by $v$ ).

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