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

DOI:
https://doi.org/10.1090/S0002-9947-01-02827-6

Published electronically:
March 16, 2001

MathSciNet review:
1828461

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Abstract | References | Similar Articles | Additional Information

Let 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 as ``lifts'' of relative discrete series characters on smaller groups called two-structure groups for . 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 or . They are not necessarily subgroups of , but they ``share'' the relatively compact Cartan subgroup and certain other Cartan subgroups with . 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.

**1.**Harish-Chandra,*Discrete series for semisimple Lie groups I*, Acta Math.,**113**(1965), 241-318. MR**36:2744****2.**Harish-Chandra,*Harmonic analysis on real reductive groups I*, J. Funct. Anal.,**19**(1975), 104-204. MR**53:3201****3.**R. Herb,*Fourier inversion and the Plancherel theorem*, (Proc. Marseille Conf. 1980), Lecture Notes in Math. Vol. 880, Springer-Verlag, Berlin and New York, 1981, 197-210. MR**83f:22013****4.**R. Herb,*The Plancherel theorem for semisimple Lie groups without compact Cartan subgroups*, (Proc. Marseille Conf. 1982), Lecture Notes in Math. Vol. 1020, Springer-Verlag, Berlin and New York, 1983, 73-79. MR**85a:22001****5.**R. Herb,*Discrete series characters and two-structures*, Trans. AMS,**350**(1998), 3341-3369. MR**98k:22058****6.**R. Herb,*Stable discrete series characters as lifts from complex two-structure groups*, Pacific J. Math.**196**(2000), 187-212. CMP**2001:04****7.**R. Herb,*Two-structures and discrete series character formulas*, Proceedings of Symposia in Pure Mathematics, Vol. 68, A.M.S. Providence RI, 2000, 285-319. CMP**2000:16****8.**R. Herb and J.A. Wolf,*The Plancherel theorem for general semisimple groups*, Compositio Math.,**57**(1986), 271-355. MR**87h:22020****9.**A.W. Knapp,*Representation Theory of Semisimple Groups, An Overview Based on Examples*, Princeton U. Press, Princeton, N.J., 1986. MR**87j:22022****10.**D. Shelstad,*Orbital integrals and a family of groups attached to a real reductive group*, Ann. Sci. Ecole Norm. Sup.**12**(1979), 1-31. MR**81k:22014****11.**D. Shelstad,*Embeddings of -groups*, Canad. J. Math.**33**(1981), 513-558. MR**83e:22022****12.**D. Shelstad,*-indistinguishability for real groups*, Math. Ann.**259**(1982), 385-430. MR**84c:22017****13.**J.A. Wolf,*Unitary representations on partially holomorphic cohomology spaces*, Memoirs AMS,**138**(1974). MR**56:3194**

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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:
https://doi.org/10.1090/S0002-9947-01-02827-6

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