  In this paper we study general hyperplane sections of adjoint and coadjoint
varieties. We show that these are the only sections of homogeneous varieties
such that a maximal torus of the automorphism group of the ambient variety
stabilizes them. We then study their geometry, provide formulas for their
classical cohomology rings in terms of Schubert classes and compute the quantum
Chevalley formula. This allows us to obtain results about the semi-simplicity
of the (small) quantum cohomology, analogous to those holding for (co)adjoint
varieties.
