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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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On certain small representations of indefinite orthogonal groups
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by Chen-bo Zhu and Jing-Song Huang PDF
Represent. Theory 1 (1997), 190-206 Request permission

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), 245โ€“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
  • 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