Enhanced Dynkin diagrams and Weyl orbits

The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of Σ is, in general, not large enough to contain the diagrams of all subsystems of Σ, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow us to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in Σ. In this paper we consider only ADE root systems (i.e., systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.