Stable Discrete Series Characters as Lifts from Complex Two-structure Groups

Let G ⊂ GC be a connected reductive linear Lie group with a Cartan subgroup B which is compact modulo the center of G. Then G has discrete series representations. Further, since G is linear the characters of discrete series representations can be averaged over the Weyl group to obtain stable discrete series characters which are constant on orbits of GC in G′, and can be regarded as the restrictions of certain class functions on the regular set GC of GC. The main theorem of this paper expresses these class functions on GC as “lifts” of analogous class functions on two-structure groups for GC. These are connected reductive complex Lie groups which are not necessarily subgroups of GC, but which “share” the Cartan subgroup BC with GC. Further, all of their simple factors have root systems of type A1 or B2 C2.