The Dynkin diagram $R$-group

Abstract:

We define an abelian group from the Dynkin diagram of a split real linear Lie group with abelian Cartan subgroups, $G$, and show that the $R_{\delta , 0}$-groups defined by Knapp and Stein are subgroups of it. The proof relies on Vogan’s approach to the $R$-groups. The $R$-group of a Dynkin diagram is easily computed just by looking at the diagram, and so it gives, for instance, quick proofs of the fact that the principal series with zero infinitesimal character of the split groups $E_6$, $E_8$, $G_2$ or $F_4$ are irreducible. The Dynkin diagram subgroup also implicitly describes a small Levi subgroup, which we hope might play a role in computing regular functions on principal nilpotent orbits. We present in the end a conjecture and some evidence in this direction.