A Geometric Construction of Crystal Graphs Using Quiver Varieties: Extension to the Non-simply Laced Case

We consider a generalization of the quiver varieties of Lusztig and Nakajima to the case of all symmetrizable Kac-Moody Lie algebras. To deal with the non-simply laced case one considers admissible automorphisms of a quiver and the irreducible components of the quiver varieties fixed by this automorphism. We define a crystal structure on these irreducible components and show that the crystals obtained are isomorphic to those associated to the crystal bases of the lower half of the universal enveloping algebra and the irreducible highest weight representations of the non-simply laced Kac-Moody Lie algebra. As an application, we realize the crystal of the spin representation of so2n+1 on the set of self-conjugate Young diagrams that fit inside an n× n box.