Crystal Bases and Diagram Automorphisms

Abstract

We prove that the action of an $\omega$-root operator on the set of all paths fixed by a diagram automorphism $\omega$ of a Kac–Moody algebra $\mathfrak{g}$ can be identified with the action of a root operator for the orbit Lie algebra $\breve{\mathfrak{g}}$. Moreover, we prove that there exists a canonical bijection between the elements of the crystal base $\mathcal{B}(\infty)$ for $\mathfrak{g}$ fixed by $\omega$ and the elements of the crystal base $\breve{\mathcal{B}}(\infty)$ for $\breve{\mathfrak{g}}$. Using this result, we give twining character formulas for the "negative part" of the quantized universal enveloping algebra $U_q(\mathfrak{g})$ and for certain modules of Demazure type.