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.
