## Nonvanishing of a certain sesquilinear form in the theta correspondence

HTML articles powered by AMS MathViewer

- by Hongyu He PDF
- Represent. Theory
**5**(2001), 437-454 Request permission

## Abstract:

Suppose $2n+1 \geq p+q$. In an earlier paper in 2000 we study a certain sesquilinear form $(,)_{\pi }$ introduced by Jian-Shu Li in 1989. For $\pi$ in the semistable range of $\theta (MO(p,q) \rightarrow MSp_{2n}(\mathbb {R}))$, if $(,)_{\pi }$ does not vanish, then it induces a sesquilinear form on $\theta (\pi )$. In another work in 2000 we proved that $(,)_{\pi }$ is positive semidefinite under a mild growth condition on the matrix coefficients of $\pi$. In this paper, we show that either $(,)_{\pi }$ or $(,)_{\pi \otimes \det }$ is nonvanishing. These results combined with one result of Przebinda suggest the existence of certain unipotent representations of $Mp_{2n}(\mathbb {R})$ beyond unitary representations of low rank.## References

- V. Bargmann,
*On a Hilbert space of analytic functions and an associated integral transform*, Comm. Pure Appl. Math.**14**(1961), 187–214. MR**157250**, DOI 10.1002/cpa.3160140303 - R. H. J. Germay,
*Généralisation de l’équation de Hesse*, Ann. Soc. Sci. Bruxelles Sér. I**59**(1939), 139–144 (French). MR**86**, DOI 10.1007/978-3-662-12918-0 - Sigurđur Helgason,
*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455** - R. H. J. Germay,
*Généralisation de l’équation de Hesse*, Ann. Soc. Sci. Bruxelles Sér. I**59**(1939), 139–144 (French). MR**86**, DOI 10.1090/surv/083 - Sigurdur Helgason,
*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561** - Hongyu L. He,
*An analytic compactification of symplectic group*, J. Differential Geom.**51**(1999), no. 2, 375–399. MR**1728304**, DOI 10.4310/jdg/1214425071
hhe1 Hongyu L. He, Compactification of Classical Groups, to appear in - Hongyu He,
*Theta correspondence. I. Semistable range: construction and irreducibility*, Commun. Contemp. Math.**2**(2000), no. 2, 255–283. MR**1759791**, DOI 10.1142/S0219199700000128
unit Hongyu L. He, Unitary Representations and Theta Correspondence: From Orthogonal Groups to Symplectic Groups, submitted to - Roger Howe,
*Transcending classical invariant theory*, J. Amer. Math. Soc.**2**(1989), no. 3, 535–552. MR**985172**, DOI 10.1090/S0894-0347-1989-0985172-6 - Roger Howe,
*Small unitary representations of classical groups*, Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, Calif., 1984) Math. Sci. Res. Inst. Publ., vol. 6, Springer, New York, 1987, pp. 121–150. MR**880374**, DOI 10.1007/978-1-4612-4722-7_{4} - Anthony W. Knapp,
*Representation theory of semisimple groups*, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR**855239**, DOI 10.1515/9781400883974 - Shrawan Kumar,
*Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture*, Invent. Math.**93**(1988), no. 1, 117–130. MR**943925**, DOI 10.1007/BF01393689 - M. Kashiwara and M. Vergne,
*On the Segal-Shale-Weil representations and harmonic polynomials*, Invent. Math.**44**(1978), no. 1, 1–47. MR**463359**, DOI 10.1007/BF01389900 - 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** - Jian-Shu Li,
*Singular unitary representations of classical groups*, Invent. Math.**97**(1989), no. 2, 237–255. MR**1001840**, DOI 10.1007/BF01389041 - Jian-Shu Li,
*Theta lifting for unitary representations with nonzero cohomology*, Duke Math. J.**61**(1990), no. 3, 913–937. MR**1084465**, DOI 10.1215/S0012-7094-90-06135-6 - Tomasz Przebinda,
*Characters, dual pairs, and unitary representations*, Duke Math. J.**69**(1993), no. 3, 547–592. MR**1208811**, DOI 10.1215/S0012-7094-93-06923-2 - P. L. Robinson and J. H. Rawnsley,
*The metaplectic representation, $\textrm {Mp}^c$ structures and geometric quantization*, Mem. Amer. Math. Soc.**81**(1989), no. 410, iv+92. MR**1015418**, DOI 10.1090/memo/0410 - David A. Vogan Jr.,
*Unitary representations of reductive Lie groups*, Annals of Mathematics Studies, vol. 118, Princeton University Press, Princeton, NJ, 1987. MR**908078**, DOI 10.1515/9781400882380

*Communications in Analysis and Geometry*, 2000.

*Journal of Functional Analysis*, 2000. thesis Hongyu L. He,

*Howe’s Rank and Dual Pair Correspondence in Semistable Range*, M.I.T. Thesis, (1-127).

## Additional Information

**Hongyu He**- Affiliation: Department of Mathematics and Statistics, Georgia State University, Atlanta, Georgia 30303-3083
- Email: matjnl@livingstone.cs.gsu.edu
- Received by editor(s): April 24, 2001
- Received by editor(s) in revised form: July 30, 2001
- Published electronically: October 30, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Represent. Theory
**5**(2001), 437-454 - MSC (2000): Primary 22E45
- DOI: https://doi.org/10.1090/S1088-4165-01-00140-6
- MathSciNet review: 1870597