Discrete Series Characters as Lifts from Two-structure Groups

Abstract

Let G 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 G as “lifts” of relative discrete series characters on smaller groups called two-structure groups for G 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 S L ( 2 , C ) SL(2, \mathbf {C}) or S O ( 5 , C ) SO(5, \mathbf {C} ) . They are not necessarily subgroups of G G , but they “share” the relatively compact Cartan subgroup and certain other Cartan subgroups with G 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.