Three kinds of extremal weight vectors fixed by a diagram automorphism

We define actions of a diagram automorphism on certain crystals, and then study three kinds of “extremal” elements of them that are fixed by a diagram automorphism. We will show that such extremal elements can be identified with extremal elements of the corresponding crystal for the orbit Lie algebra. As a consequence, we prove that there exists a canonical injection from the elements of the crystal base of an extremal weight module fixed by a diagram automorphism into the elements of the crystal base of the corresponding extremal weight module for the orbit Lie algebra.