Cyclotomic Discriminantal Arrangements and Diagram Automorphisms of Lie Algebras

Abstract

Abstract We identify a class of affine hyperplane arrangements that we call cyclotomic discriminantal arrangements. We establish correspondences between the flag and Aomoto complexes of such arrangements and chain complexes for nilpotent subalgebras of Kac–Moody type Lie algebras with diagram automorphisms. As part of this construction, we find that flag complexes naturally give rise to a certain cocycle on the fixed-point subalgebras of such diagram automorphisms. As a byproduct, we show that the Bethe vectors of cyclotomic Gaudin models associated to diagram automorphisms are nonzero. We also obtain the Poincare polynomial for the cyclotomic discriminantal arrangements.