Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A cocycle formula for the quaternionic discrete series

Author: Robert W. Donley Jr.
Journal: Proc. Amer. Math. Soc. 131 (2003), 1943-1951
MSC (2000): Primary 22E46, 53C65
Published electronically: November 14, 2002
MathSciNet review: 1955285
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Schmid's proof of the Kostant-Langlands conjecture for discrete series representations of a semisimple Lie group provides a Hilbert space realization of such representations in $L^{2}$-cohomology. We give an explicit description of these harmonic forms for the quaternionic discrete series.

References [Enhancements On Off] (What's this?)

  • [Al] D. V. Alekseevskii, Compact quaternion spaces, Functional Anal. and Its Appl. 2 (1968), 106-114. MR 37:6869 (Russian)
  • [Ba] L. Barchini, Strongly harmonic forms for representations in the discrete series, J. Funct. Anal. 161 (1999), 111-131. MR 2000a:22019
  • [Do] R. W. Donley, Jr., Orthogonality relations and harmonic forms for semisimple Lie groups, J. Funct. Anal. 170 (2000), 141-160. MR 2001j:22016
  • [Go] D. Gordon, Quaternionic discrete series, Represent. Theory 3 (1999), 32-57. MR 2001g:22025
  • [GS] P. Griffiths, and W. Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969), 253-302. MR 41:4587
  • [GW] B. H. Gross, and N. R. Wallach, On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math. 481 (1996), 73-123. MR 98f:22022
  • [HC] Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1-111. MR 36:2745
  • [He] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. MR 26:2986
  • [Kn] A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser, Boston, 1996. MR 98b:22002
  • [Sc] W. Schmid, $L^{2}$-cohomology and the discrete series, Ann. of Math. 103 (1976), 375-394. MR 53:716
  • [Wo] J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. and Mech. 14 (1965), 1033-1047. MR 32:3020

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 22E46, 53C65

Retrieve articles in all journals with MSC (2000): 22E46, 53C65

Additional Information

Robert W. Donley Jr.
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203

Keywords: Quaternionic discrete series, harmonic forms, matrix coefficients
Received by editor(s): January 22, 2002
Published electronically: November 14, 2002
Additional Notes: The author was supported by MSRI. Research at MSRI was supported in part by NSF grant DMS-9810361
Communicated by: Rebecca Herb
Article copyright: © Copyright 2002 American Mathematical Society