Extremal vectors are defined in Kashiwara's paper. The definition is as follows.
Simple reflections in the Weyl group of $\mathfrak{g}$ acts on the crystal basis of integrable $U_q(\mathfrak{g})$-modules as \begin{align} S_i(b) = \begin{cases} \tilde{f}_i^{\langle h_i, wt(b) \rangle}b, & \langle h_i, wt(b) \rangle \geq 0, \\ \tilde{e}_i^{-\langle h_i, wt(b) \rangle}b, & \langle h_i, wt(b) \rangle \leq 0. \end{cases} \end{align} An vector $b$ in the crystal basis is called an extremal vector if for any $w \in W$, any $i \in I$, $S_w b$ is killed by $\tilde{e}_i$ or $\tilde{f}_i$. What is the intuition of the definition of extremal vector? Is a highest weight vector an extremal vector? Thank you very much.
