Quivers and the cohomology of homogeneous vector bundles

Abstract

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE-type as the cohomology of a complex explicitly described. The main tool is the equivalence (introduced by Bondal, Kapranov, and Hille) between the category of homogeneous bundles and the category of representations of a certain quiver ${\mathcal{Q}}_X$ with relations. We prove that the relations are the commutative ones on projective spaces, but they involve additional scalars on general Grassmannians. In addition, we introduce moduli spaces of homogeneous bundles