Abstract
In this paper, we will give a “natural” description of a certain subset of the unitary dual of Sp2n(ℝ), the real symplectic group in 2n variables. The description will be in terms of the unitary duals of orthogonal groups. The representations of the symplectic group which we describe here are “small” in a well-defined sense explained below. The description relies heavily on the Mackey theory of induced representations, and on the theory of the oscillator representation. This paper is essentially a continuation of ↑H←. Results similar to those described here are valid for other classical Lie groups, and for classical groups over p-adic fields.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. Howe, On a notion of rank for unitary representation of the classical groups, Harmonic Analysis and Group Representations, C.I.M.E. II Ciclo 1980, Cortona-Arezzo, A. Figà-Talamanca coord., Liquori Editore, Naples, 1982, 223–331.
R. Howe, On the role of the Heisenberg group in harmonic analysis, B.A.M.S. (New Series) v.3(1980), 821–843.
G. Mackey, Infinite dimensional group representations, B.A.M.S., v.69(1963), 628–686.
G. Mackey, Unitary representations of group extensions I, Acta Math., v.99(1958), 265–311.
G. Mackey, Induced representations of locally compact groups II, Ann. of Math., v.58(1953), 193–221.
G. Mackey, The Theory of Unitary Representations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago and London, 1976.
R. Rao, On some explicit formulas in the theory of Weil representation, preprint.
M. Rieffel, Induced representations of C* –algebras, Adv. Math., 18(1979), 176–257.
M. Saito, Representations unitaires des groupes symplectiques, J. Math. Soc. Japan, 24(1972), 232–251.
R. Scaramuzzi, Yale Doctoral Dissertation, 1985.
D. Shale, Linear symmetries of free boson fields, T.A.M.S., 103(1962), 149–167.
J. Tits, Groups semi-simples isotropes, Colloque sur la theorie des groupes algebriques (Brussels, 1962), Gauthier-Villars, Paris, 1962, 137–147.
A. Weil, Sur Certains groupes d’operateurs unitaires, Acta Math. III, (1964), 143–211.
A. Weil, Sur la formula de Siegel dans la theorie des groupes classiques, Acta Math. 113(1965), 1–87.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag New York Inc.
About this paper
Cite this paper
Howe, R. (1987). Small Unitary Representations Of Classical Groups. In: Moore, C.C. (eds) Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics. Mathematical Sciences Research Institute Publications, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4722-7_4
Download citation
DOI: https://doi.org/10.1007/978-1-4612-4722-7_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-9130-5
Online ISBN: 978-1-4612-4722-7
eBook Packages: Springer Book Archive