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Representation Theory

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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
Published electronically: July 17, 1997
MathSciNet review: 1457244
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Abstract: For any $n\in \mathbb {N}$ such that $2n\leq \min (p,q)$, we construct a representation $\pi _{n}$ of $O(p,q)$ with $p+q$ even as the kernel of a commuting set of $\frac {n(n+1)}{2}$ number of $O(p,q)$-invariant differential operators in the space of $C^{\infty }$ functions on an isotropic cone with a distinguished $GL_{n}(\mathbb {R})$-homogeneity degree. By identifying $\pi _{n}$ with a certain representation constructed via the formalism of the theta correspondence, we show (except when $p=q=2n$) that the space of $K$-finite vectors of $\pi _{n}$ is the $(\mathfrak {g},K)$-module of an irreducible unitary representation of $O(p,q)$ with Gelfand-Kirillov dimension $n(p+q-2n-1)$. Our construction generalizes the work of Binegar and Zierau (Unitarization of a singular representation of $SO_{e}(p,q)$, Commun. Math. Phys. 138 (1991), 245-258) for $n=1$.

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Additional Information

Chen-bo Zhu
Affiliation: Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 119260

Jing-Song Huang
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

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