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

Full-text PDF Free Access

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. Construction of invariant eigendistributions*, Acta Math.**113**(1965), 241–318. MR**0219665****2.**Harish-Chandra,*Harmonic analysis on real reductive groups. I. The theory of the constant term*, J. Functional Analysis**19**(1975), 104–204. MR**0399356****3.**Rebecca A. Herb,*Fourier inversion and the Plancherel theorem*, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp. 197–210. MR**644834****4.**J. Carmona and M. Vergne (eds.),*Noncommutative harmonic analysis and Lie groups*, Lecture Notes in Mathematics, vol. 1020, Springer-Verlag, Berlin, 1983. MR**733457****5.**Rebecca A. Herb,*Discrete series characters and two-structures*, Trans. Amer. Math. Soc.**350**(1998), no. 8, 3341–3369. MR**1422607**, 10.1090/S0002-9947-98-01958-8**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.**Rebecca A. Herb and Joseph A. Wolf,*The Plancherel theorem for general semisimple groups*, Compositio Math.**57**(1986), no. 3, 271–355. MR**829325****9.**Anthony W. Knapp,*Representation theory of semisimple groups*, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR**855239****10.**Diana Shelstad,*Orbital integrals and a family of groups attached to a real reductive group*, Ann. Sci. École Norm. Sup. (4)**12**(1979), no. 1, 1–31. MR**532374****11.**D. Shelstad,*Embeddings of 𝐿-groups*, Canad. J. Math.**33**(1981), no. 3, 513–558. MR**627641**, 10.4153/CJM-1981-044-4**12.**D. Shelstad,*𝐿-indistinguishability for real groups*, Math. Ann.**259**(1982), no. 3, 385–430. MR**661206**, 10.1007/BF01456950**13.**Joseph A. Wolf,*Unitary representations of maximal parabolic subgroups of the classical groups*, Mem. Amer. Math. Soc.**8**(1976), no. 180, iii+193. MR**0444847**

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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

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