## On certain small representations of indefinite orthogonal groups

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- by Chen-bo Zhu and Jing-Song Huang
- Represent. Theory
**1**(1997), 190-206 - DOI: https://doi.org/10.1090/S1088-4165-97-00031-9
- Published electronically: July 17, 1997
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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), 245โ258) for $n=1$.

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## Bibliographic Information

**Chen-bo Zhu**- Affiliation: Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 119260
- MR Author ID: 305157
- ORCID: 0000-0003-3819-1458
- Email: matzhucb@leonis.nus.sg
**Jing-Song Huang**- Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
- MR Author ID: 304754
- Email: mahuang@uxmail.ust.hk
- Received by editor(s): September 4, 1996
- Received by editor(s) in revised form: January 9, 1997
- Published electronically: July 17, 1997
- © Copyright 1997 American Mathematical Society
- Journal: Represent. Theory
**1**(1997), 190-206 - MSC (1991): Primary 22E45, 22E46
- DOI: https://doi.org/10.1090/S1088-4165-97-00031-9
- MathSciNet review: 1457244