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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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  • [A] J. Adams, The theta correspondence over $\mathbb {R}$, Preprint, Workshop at the University of Maryland (1994).
  • [BK] R. Brylinski and B. Kostant, Minimal representations, geometric quantization, and unitarity, Proc. Natl. Acad. Sci. USA 91 (1994), 6026-6029. MR 95d:58059
  • [BZ] B. Binegar and R. Zierau, Unitarization of a singular representation of $SO_{e}(p,q)$, Commun. Math. Phys. 138 (1991), 245-258. MR 92h:22027
  • [CM] D. Collingwood and W. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, New York, 1993. MR 94j:17001
  • [H1] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570. MR 90h:22015a
  • [H2] R. Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), 535-552. MR 90k:22016
  • [H3] R. Howe, Dual pairs in physics: Harmonic oscillators, photons, electrons, and singletons, Lectures in Appl. Math., Vol. 21, Amer. Math. Soc., Providence, R.I. (1985), 179-206. MR 86i:22036
  • [H4] R. Howe, $\theta $-series and invariant theory, Proc. Sympos. Pure Math., Vol. 33, Part 1, Automorphic forms, representations and L-functions (1979), 275-286. MR 81f:22034
  • [H5] R. Howe, A notion of rank for unitary representations of classical groups, C.I.M.E. Summer School on Harmonic Analysis, Cortona 1980.
  • [HT] R. Howe and E.-C. Tan, Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series, Bull. Amer. Math. Soc. 28 (1993), 1-74. MR 93j:22027
  • [K1] B. Kostant, The principle of triality and a distinguished representation of $SO(4,4)$, Differential geometric methods in theoretical physics. Bleuler, K., Werner, M. (eds.) Series C: Math. and Phys., Sci., Vol. 250. MR 90h:22016
  • [K2] B. Kostant, The Vanishing of Scalar Curvature and the Minimal Representation of $SO(4,4)$, Operator algebras, Unitary representations, Enveloping algebras and Invariant theory, Proceedings of the Colloque en l'Honneur de Jacques Dixmier, 1989. MR 92g:22031
  • [Ku] S. Kudla, Seesaw dual reductive pairs, Progr. Math. 46 (1983), 244-268. MR 86b:22032
  • [KR1] S. Kudla and S. Rallis, Degenerate principal series and invariant distributions, Israel J. Math. 69 (1990), 25-45. MR 91e:22016
  • [KR2] S. Kudla and S. Rallis, Ramified degenerate principal series representations for $Sp(n)$, Israel J. Math 78 (1992), 209-256. MR 94a:22035
  • [KV] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), 1-47. MR 57:3311
  • [L1] J. S. Li, Singular unitary representations of classical groups, Invent. Math. 97 (1989), 237-255. MR 90h:22021
  • [L2] J. S. Li, On the classification of irreducible low rank unitary representations of classical groups, Compositio Mathematica 71 (1989), 29-48. MR 90k:22027
  • [LZ1] S. T. Lee and C. B. Zhu, Degenerate principal series and local theta correspondence, Trans. Amer. Math. Soc. (to appear).
  • [LZ2] S. T. Lee and C. B. Zhu, Degenerate principal series and local theta correspondence II, Israel Jour. Math. (to appear).
  • [M] W. McGovern, Rings of regular functions on nilpotent orbits II: Model algebras and orbits, Commun. in Algebra 22 (1994), 765-772. MR 95b:22035
  • [S] S. Sahi, Explicit Hilbert spaces for certain unipotent representations, Invent. Math. 110 (2) (1992), 409-418. MR 93i:22016
  • [T] T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc. 219 (1976), 1-46. MR 53:3210
  • [V] D. Vogan, Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75-98. MR 58:22205
  • [W] H. Weyl, The classical groups, Princeton University Press, Princeton, New Jersey, 1939. MR 1:42c
  • [Z] C. B. Zhu, Invariant distributions of classical groups, Duke Math. Jour. 65 (1) (1992), 85-119. MR 92k:22022

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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

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