Discrete Series Characters as Lifts from Two-structure Groups

Let $G$ be a connected reductive Lie group with a relatively compact Cartan subgroup. Then it has relative discrete series representations. The main result of this paper is a formula expressing relative discrete series characters on $G$ as ``lifts'' of relative discrete series characters on smaller groups called two-structure groups for $G$. The two-structure groups are connected reductive Lie groups which are locally isomorphic to the direct product of an abelian group and simple groups which are real forms of $SL(2, \mathbf{C})$ or $SO(5, { {\bf C} })$. They are not necessarily subgroups of $G$, but they ``share'' the relatively compact Cartan subgroup and certain other Cartan subgroups with $G$. The character identity is similar to formulas coming from endoscopic lifting, but the two-structure groups are not necessarily endoscopic groups, and the characters lifted are not stable. Finally, the formulas are valid for non-linear as well as linear groups.