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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Discrete Series Characters as Lifts from Two-structure Groups


Author: Rebecca A. Herb
Journal: Trans. Amer. Math. Soc. 353 (2001), 2557-2599
MSC (2000): Primary 22E30, 22E45
Published electronically: March 16, 2001
MathSciNet review: 1828461
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Abstract:

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.


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Additional Information

Rebecca A. Herb
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: rah@math.umd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02827-6
PII: S 0002-9947(01)02827-6
Received by editor(s): June 1, 1999
Published electronically: March 16, 2001
Additional Notes: Supported in part by NSF Grant DMS 9705645
Article copyright: © Copyright 2001 American Mathematical Society