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

Full-text PDF Free Access

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 .

**[A]**J. Adams,*The theta correspondence over*, Preprint, Workshop at the University of Maryland (1994).**[BK]**Ranee Brylinski and Bertram Kostant,*Minimal representations, geometric quantization, and unitarity*, Proc. Nat. Acad. Sci. U.S.A.**91**(1994), no. 13, 6026–6029. MR**1278630**, https://doi.org/10.1073/pnas.91.13.6026**[BZ]**B. Binegar and R. Zierau,*Unitarization of a singular representation of 𝑆𝑂(𝑝,𝑞)*, Comm. Math. Phys.**138**(1991), no. 2, 245–258. MR**1108044****[CM]**David H. Collingwood and William M. McGovern,*Nilpotent orbits in semisimple Lie algebras*, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR**1251060****[H1]**Roger Howe,*Remarks on classical invariant theory*, Trans. Amer. Math. Soc.**313**(1989), no. 2, 539–570. MR**986027**, https://doi.org/10.1090/S0002-9947-1989-0986027-X**[H2]**Roger Howe,*Transcending classical invariant theory*, J. Amer. Math. Soc.**2**(1989), no. 3, 535–552. MR**985172**, https://doi.org/10.1090/S0894-0347-1989-0985172-6**[H3]**Roger Howe,*Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons*, Applications of group theory in physics and mathematical physics (Chicago, 1982) Lectures in Appl. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1985, pp. 179–207. MR**789290****[H4]**R. Howe,*𝜃-series and invariant theory*, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 275–285. MR**546602****[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]**Roger E. Howe and Eng-Chye Tan,*Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations*, Bull. Amer. Math. Soc. (N.S.)**28**(1993), no. 1, 1–74. MR**1172839**, https://doi.org/10.1090/S0273-0979-1993-00360-4**[K1]**Bertram Kostant,*The principle of triality and a distinguished unitary representation of 𝑆𝑂(4,4)*, Differential geometrical methods in theoretical physics (Como, 1987) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 250, Kluwer Acad. Publ., Dordrecht, 1988, pp. 65–108. MR**981373****[K2]**Bertram Kostant,*The vanishing of scalar curvature and the minimal representation of 𝑆𝑂(4,4)*, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 85–124. MR**1103588****[Ku]**Stephen S. Kudla,*Seesaw dual reductive pairs*, Automorphic forms of several variables (Katata, 1983) Progr. Math., vol. 46, Birkhäuser Boston, Boston, MA, 1984, pp. 244–268. MR**763017****[KR1]**Stephen S. Kudla and Stephen Rallis,*Degenerate principal series and invariant distributions*, Israel J. Math.**69**(1990), no. 1, 25–45. MR**1046171**, https://doi.org/10.1007/BF02764727**[KR2]**Stephen S. Kudla and Stephen Rallis,*Ramified degenerate principal series representations for 𝑆𝑝(𝑛)*, Israel J. Math.**78**(1992), no. 2-3, 209–256. MR**1194967**, https://doi.org/10.1007/BF02808058**[KV]**M. Kashiwara and M. Vergne,*On the Segal-Shale-Weil representations and harmonic polynomials*, Invent. Math.**44**(1978), no. 1, 1–47. MR**463359**, https://doi.org/10.1007/BF01389900**[L1]**Jian-Shu Li,*Singular unitary representations of classical groups*, Invent. Math.**97**(1989), no. 2, 237–255. MR**1001840**, https://doi.org/10.1007/BF01389041**[L2]**Jian-Shu Li,*On the classification of irreducible low rank unitary representations of classical groups*, Compositio Math.**71**(1989), no. 1, 29–48. MR**1008803****[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]**William M. McGovern,*Rings of regular functions on nilpotent orbits. II. Model algebras and orbits*, Comm. Algebra**22**(1994), no. 3, 765–772. MR**1261003**, https://doi.org/10.1080/00927879408824874**[S]**Siddhartha Sahi,*Explicit Hilbert spaces for certain unipotent representations*, Invent. Math.**110**(1992), no. 2, 409–418. MR**1185591**, https://doi.org/10.1007/BF01231340**[T]**Tuong Ton That,*Lie group representations and harmonic polynomials of a matrix variable*, Trans. Amer. Math. Soc.**216**(1976), 1–46. MR**399366**, https://doi.org/10.1090/S0002-9947-1976-0399366-1**[V]**David A. Vogan Jr.,*Gel′fand-Kirillov dimension for Harish-Chandra modules*, Invent. Math.**48**(1978), no. 1, 75–98. MR**506503**, https://doi.org/10.1007/BF01390063**[W]**H. Weyl,*The classical groups*, Princeton University Press, Princeton, New Jersey, 1939. MR**1:42c****[Z]**Chen-bo Zhu,*Invariant distributions of classical groups*, Duke Math. J.**65**(1992), no. 1, 85–119. MR**1148986**, https://doi.org/10.1215/S0012-7094-92-06504-5

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (1991):
22E45,
22E46

Retrieve articles in all journals with MSC (1991): 22E45, 22E46

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