Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Quaternionic discrete series

Author: Derek Gordon
Journal: Represent. Theory 3 (1999), 32-57
MSC (1991): Primary 22E46
Published electronically: June 2, 1999
MathSciNet review: 1694200
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This work investigates the discrete series of linear connected semi-simple noncompact groups $G$. These are irreducible unitary representations that occur as direct summands of $L^{2}(G)$.

Harish-Chandra produced discrete series representations, now called holomorphic discrete series representations, for groups $G$ with the property that, if $K$ is a maximal compact subgroup, then $G/K$ has a complex structure such that $G$ acts holomorphically. Holomorphic discrete series are extraordinarily explicit, it being possible to determine all the elements in the space and the action by the Lie algebra of $G$.

Later Harish-Chandra parametrized the discrete series in general. His argument did not give an actual realization of the representations, but later authors found realizations in spaces defined by homology or cohomology. These realizations have the property that it is not apparent what elements are in the space and what the action of the Lie algebra $G$ is.

The point of this work is to find some intermediate ground between the holomorphic discrete series and the general discrete series, so that the intermediate cases may be used to get nontrivial insights into the internal structure of the discrete series in the general case.

The author examines the Vogan-Zuckerman realization of discrete series by means of cohomological induction. An explicit complex for computing the homology on the level of a $K$ module was already known. Also, Duflo and Vergne had given information about how to compute the action of the Lie algebra of $G$.

The holomorphic discrete series are exactly those cases where the representations can be realized in homology of degree 0. The intermediate cases that are studied are those where the representation can be realized in homology of degree 1. Many of the intermediate cases correspond to the situation where $G/K$ has a quaternionic structure. The author obtains general results for ${\text{A}_{\mathfrak{q}}(\lambda )}$ discrete series in the intermediate case.

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

  • [Bar] V. Bargmann, Irreducible unitary representations for the Lorentz group, Ann. of Math. 48 (1947), 568-640. MR 9:133a
  • [Bes] A. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987. MR 88f:53087
  • [BS-K] M. W. Baldoni-Silva and A. W. Knapp, Intertwining operators and unitary representations. I, J. Funct. Anal. 82 (1989), 151-236. MR 90g:22021
  • [Che] C. Chevalley, Theory of Lie Groups, Princeton University Press, Princeton, 1946. MR 7:412c
  • [Dix] J. Dixmier, Représentations intégrables du groupe de De Sitter, Bull. Soc. Math. France. 89 (1961), 9-41. MR 25:3384
  • [D-V] M. Duflo and M. Vergne, Sur le foncteur de Zuckerman, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 467-469. MR 89h:22025
  • [EW1] T. J. Enright, R. Parthasarathy, N. R. Wallach, and J. A. Wolf, Classes of unitarizable derived functor modules, Proc. Nat. Acad. Sci. USA 80 (1983), 7047-7050.
  • [EW2] T. J. Enright, R. Parthasarathy, N. R. Wallach, and J. A. Wolf, Unitary derived functor modules with small spectrum, Acta Math. 154 (1985), 105-136. MR 86j:22026
  • [E-V] T. J. Enright and V. S. Varadarajan, On an infinitesimal characterization of the discrete series, Ann. of Math. 102 (1975), 1-15. MR 57:16472
  • [F1] R. Fueter, Die Funktionentheorie der Differentialgleichungen $\Delta u=0$ und $\Delta \Delta u=0$ mit vier reelen Variablen, Comment. Math. Helv. 7 (1935), 307-330.
  • [F2] R. Fueter, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 8 (1936), 371-378.
  • [F3] R. Fueter, Die Singularitäten der eindeutigen regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 9 (1937), 320-335.
  • [FJ1] M. Flensted-Jensen, Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), 106-146. MR 80f:43022
  • [FJ2] M. Flensted-Jensen, Discrete series for semisimple symmetric spaces, Ann. of Math. 111 (1980), 253-311. MR 81h:22015
  • [Gor] D. Gordon, Quaternionic Discrete Series of Semisimple Lie Groups, Ph.D. Thesis, State University of New York, Stony Brook, 1995.
  • [HC1] Harish-Chandra, Representations of semisimple Lie groups. IV, Amer. J. Math. 77 (1955), 743-777. MR 17:282c
  • [HC2] Harish-Chandra, Representations of semisimple Lie groups. V, Amer. J. Math. 78 (1956), 1-41. MR 18:490c
  • [HC3] Harish-Chandra, Representations of semisimple Lie groups. VI, Amer. J. Math. 78 (1956), 564-628. MR 18:490d
  • [HC4] Harish-Chandra, Discrete series for semisimple Lie groups. II, Explicit determination of the characters, Acta Math. 116 (1966), 1-111. MR 36:2745
  • [Hel] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. MR 80k:53081
  • [H-K] K. Hoffman and R. Kunze, Linear Algebra, Second Edition, Prentice-Hall, Englewood Cliffs, N.J., 1971. MR 43:1998
  • [H-R] R. Hotta and R. Parthasarathy, Multiplicity formulae for discrete series, Invent. Math. 26 (1974), 133-178. MR 50:539
  • [Hum] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. MR 48:2197
  • [K1] A. W. Knapp, Introduction to Representations in Analytic Cohomology, Baston-Eastwood Summer Research Conference, 1992. MR 94k:22031
  • [K2] A. W. Knapp, Lie Groups, Lie Algebras, and Cohomology, Mathematical Notes No. 34, Princeton University Press, Princeton 1988. MR 89j:22034
  • [K3] A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, Princeton, 1986. MR 87j:22022
  • [K4] A. W. Knapp, Bounded symmetric domains and holomorphic discrete series, Symmetric Spaces ( W.M. Boothby and G.L. Weiss, eds.), Marcel Dekker, New York, 1972, 211-246. MR 57:537
  • [K-V] A. W. Knapp and D. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, 1995. MR 96c:22023
  • [K-W] A. W. Knapp and N. R. Wallach, Szegö kernels associated with discrete series, Invent. Math., 34 (1976), 163-200. MR 82i:22016; MR 54:7704
  • [Lan] R. P. Langlands, Dimension of spaces of automorphic forms, Algebraic groups and Discontinuous Subgroups, Proc. Symp. in Pure Math. 9 (1966), American Mathematical Society, Providence, R.I., 235-257. MR 35:3010
  • [Rud] W. Rudin, Functional analysis, Second Edition, McGraw-Hill, New York, 1991. MR 92k:46001
  • [S1] W. Schmid, Homogeneous complex manifolds and representations of semi-simple Lie groups, Ph.D. dissertation, University of California, Berkeley, 1967, in ``Representation Theory and Harmonic Analysis on Semisimple Lie Groups", Math. Surveys and Monographs, American Mathematical Society, Providence, (1989), 223-286. MR 90i:22025
  • [S2] W. Schmid, On the realization of the discrete series of a semisimple Lie group, Rice University Studies, 56 No. 2, (1970), 99-108. MR 43:3401
  • [S3] W. Schmid, $L^{2}$-cohomology and the discrete series, Ann. of Math., 103 (1976), 375-394. MR 53:716
  • [Sud] A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Phil. Soc., 85 (1979), 199-225. MR 80g:30031
  • [Tak] R. Takahashi, Sur les représentations unitaires des groupes de Lorentz Généralisés, Bull. Soc. Math. France., 91 (1963), 289-433. MR 31:3544
  • [Vog] D. Vogan, Representations of Real Reductive Lie Groups, Birkhäuser, Boston, 1981. MR 83c:22022
  • [Wol] J. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. and Mech. 14 (1965), 1033-1047. MR 32:3020
  • [Zuc] G. L. Zuckerman, Lecture Series, ``Construction of representations via derived functors'', Institute for Advanced Study, Princeton, N.J., Jan.-Mar. 1978.

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (1991): 22E46

Retrieve articles in all journals with MSC (1991): 22E46

Additional Information

Derek Gordon
Affiliation: Laboratory of Statistical Genetics, Rockefeller University, 1230 York Avenue, New York, New York 10021

Received by editor(s): February 19, 1998
Received by editor(s) in revised form: November 18, 1998
Published electronically: June 2, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society