On certain small representations of indefinite orthogonal groups

Authors:
Chen-bo Zhu and Jing-Song Huang

Journal:
Represent. Theory **1** (1997), 190-206

MSC (1991):
Primary 22E45, 22E46

DOI:
https://doi.org/10.1090/S1088-4165-97-00031-9

Published electronically:
July 17, 1997

MathSciNet review:
1457244

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Abstract | References | Similar Articles | Additional Information

Abstract: For any such that , we construct a representation of with even as the kernel of a commuting set of number of -invariant differential operators in the space of functions on an isotropic cone with a distinguished -homogeneity degree. By identifying with a certain representation constructed via the formalism of the theta correspondence, we show (except when ) that the space of -finite vectors of is the -module of an irreducible unitary representation of with Gelfand-Kirillov dimension . Our construction generalizes the work of Binegar and Zierau (*Unitarization of a singular representation of *, Commun. Math. Phys. **138** (1991), 245-258) for .

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Additional Information

**Chen-bo Zhu**

Affiliation:
Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 119260

Email:
matzhucb@leonis.nus.sg

**Jing-Song Huang**

Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

Email:
mahuang@uxmail.ust.hk

DOI:
https://doi.org/10.1090/S1088-4165-97-00031-9

Keywords:
Orthogonal groups,
isotropic cones,
theta correspondence,
Howe quotient,
Gelfand-Kirillov dimension,
nilpotent orbits

Received by editor(s):
September 4, 1996

Received by editor(s) in revised form:
January 9, 1997

Published electronically:
July 17, 1997

Article copyright:
© Copyright 1997
American Mathematical Society