## Duke Mathematical Journal

### Theta lifting of unitary lowest weight modules and their associated cycles

#### Abstract

We consider a reductive dual pair (G, G') in the stable range with G' the smaller member and of Hermitian symmetric type. We study the theta lifting of (holomorphic) nilpotent K'-orbits in relation to the theta lifting of unitary lowest weight representations of G'. We determine the associated cycles of all such representations. In particular, we prove that the multiplicity in the associated cycle is preserved under the theta lifting. We also develop a theory for the lifting of covariants arising from double fibrations by affine quotient maps.

#### Article information

Source
Duke Math. J., Volume 125, Number 3 (2004), 415-465.

Dates
First available in Project Euclid: 18 November 2004

https://projecteuclid.org/euclid.dmj/1100793677

Digital Object Identifier
doi:10.1215/S0012-7094-04-12531-X

Mathematical Reviews number (MathSciNet)
MR2166751

Zentralblatt MATH identifier
1078.22010

#### Citation

Nishiyama, Kyo; Zhu, Chen-Bo. Theta lifting of unitary lowest weight modules and their associated cycles. Duke Math. J. 125 (2004), no. 3, 415--465. doi:10.1215/S0012-7094-04-12531-X. https://projecteuclid.org/euclid.dmj/1100793677

#### References

• \lccJ. D. Adams, Discrete spectrum of the dual pair $(\emph{O}(p,q),Sp(2m))$, Invent. Math. 74 (1983), 449–475.
• ––––, “$L$-functoriality for dual pairs” in Orbites unipotentes et représentations, II, Astérisque 171172, Soc. Math. France, Montrouge, 1989, 85–129.
• \lccA. Daszkiewicz, W. Kraśkiewicz, and T. Przebinda, Dual pairs and Kostant-Sekiguchi correspondence, II, preprint, 2003, http://crystal.ou.edu/$\tilde{\ }$tprzebin/dkp2.pdf
• \lccM. G. Davidson, T. J. Enright, and R. J. Stanke, Differential operators and highest weight representations, Mem. Amer. Math. Soc. 94 (1991), no. 455.
• \lccT. Enright, R. Howe, and N. Wallach, “A classification of unitary highest weight modules” in Representation Theory of Reductive Groups (Park City, Utah, 1982), Progr. Math. 40, Birkhäuser, Boston, 1983, 97–143.
• \lccT. J. Enright and R. Parthasarathy, “A proof of a conjecture of Kashiwara and Vergne” in Noncommutative Harmonic Analysis and Lie Groups (Marseille, 1980), Lecture Notes in Math. 880, Springer, Berlin, 1981, 74–90.
• \lccK. I. Gross and R. A. Kunze, Finite-dimensional induction and new results on invariants for classical groups, I, Amer. J. Math. 106 (1984), 893–974.
• \lccHarish-Chandra, Representations of semisimple Lie groups, IV, Amer. J. Math. 77 (1955), 743–777.
• ––––, Representations of semisimple Lie groups, V, Amer. J. Math. 78 (1956), 1–41.
• \lccR. Howe, “$\theta$-series and invariant theory” in Automorphic Forms, Representations and $L$-Functions (Corvallis, Ore., 1977), Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, 1979, 275–286.
• ––––, “Reciprocity laws in the theory of dual pairs” in Representation Theory of Reductive Groups (Park City, Utah, 1982), Progr. Math. 40, Birkhäuser, Boston, 1983, 159–175.
• ––––, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539–570; Erratum, Trans. Amer. Math. Soc. 318 (1990), 823. ;
• ––––, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), 535–552.
• \lccR. Howe and C.-B. Zhu, Eigendistributions for orthogonal groups and representations of symplectic groups, J. Reine Angew. Math. 545 (2002), 121–166.
• \lccH. P. Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), 385–412.
• \lccM. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), 1–47.
• \lccA. W. Knapp, Branching theorems for compact symmetric spaces, Represent. Theory 5 (2001), 404–436.
• \lccB. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.
• \lccS. S. Kudla, “Seesaw dual reductive pairs” in Automorphic Forms of Several Variables, Progr. Math. 46, Birkhäuser, Boston, 1983, 244–268.
• \lccS. S. Kudla and S. Rallis, Degenerate principal series and invariant distributions, Israel J. Math. 69 (1990), 25–45.
• \lccS. T. Lee and C.-B. Zhu, Degenerate principal series and local theta correspondence, II, Israel J. Math. 100 (1997), 29–59.
• ––––, Degenerate principal series and local theta correspondence, Trans. Amer. Math. Soc. 350 (1998), 5017–5046.
• \lccJ.-S. Li, Singular unitary representations of classical groups, Invent. Math. 97 (1989), 237–255.
• ––––, Theta lifting for unitary representations with nonzero cohomology, Duke Math. J. 61 (1990), 913–937.
• ––––, “Theta series and construction of automorphic forms” in Representation Theory of Groups and Algebras, Contemp. Math. 145, Amer. Math. Soc., Providence, 1993, 237–248.
• \lccH. Y. Loke, Howe quotients of unitary characters and unitary lowest weight modules, preprint, 2001, http://www.math.nus.edu.sg/$\tilde{\ }$matlhy/loke.pdf
• \lccK. Nishiyama, Multiplicity-free actions and the geometry of nilpotent orbits, Math. Ann. 318 (2000), 777–793.
• ––––, “Theta lifting of two-step nilpotent orbits for the pair O$(p, q) \times$ Sp$(2n, \mathbb{R})$” in Infinite Dimensional Harmonic Analysis (Kyoto, 1999), Gräbner, Altendorf, Germany, 2000, 278–289.
• \lccK. Nishiyama, H. Ochiai, and K. Taniguchi, “Bernstein degree and associated cycles of Harish-Chandra modules–-Hermitian symmetric case” in Nilpotent Orbits, Associated Cycles and Whittaker Models for Highest Weight Representations, Astérisque 273, Soc. Math. France, Montrouge, 2001, 13–80.
• \lccK. Nishiyama, H. Ochiai, K. Taniguchi, H. Yamashita, and S. Kato, Nilpotent Orbits, Associated Cycles and Whittaker Models for Highest Weight Representations, Astérisque 273, Soc. Math. France, Montrouge, 2001.
• \lccK. Nishiyama, H. Ochiai, and C.-B. Zhu, Theta lifting of nilpotent orbits for symmetric pairs, to appear in Trans. Amer. Math. Soc., http://www.math.kyoto-u.ac.jp/$\tilde{\ }$kyo/mypapere.html
• \lccK. Nishiyama and C.-B. Zhu, Theta lifting of holomorphic discrete series: The case of U$(n, n) \times$ U$(p, q)$, Trans. Amer. Math. Soc. 353 (2001), 3327–3345.
• \lccT. Ohta, Nilpotent orbits of $Z_4$-graded Lie algebra and geometry of the moment maps associated to the dual pair (U$(p,q)$, U$(r,s)$), to appear in Publ. Res. Inst. Math. Sci.
• ––––, Nilpotent orbits of $Z_4$-graded Lie algebra and geometry of the moment maps associated to the dual pairs (O$(p,q), \Sp(2n, \mathbb{R})$) and (O$^*(2n), \Sp(p,q)$), preprint, 2001.
• \lccA. Paul and P. E. Trapa, One-dimensional representations of ${\mathrm{U}}(p,q)$ and the Howe correspondence, J. Funct. Anal. 195 (2002), 129–166.
• \lccT. Przebinda, Characters, dual pairs, and unipotent representations, J. Funct. Anal. 98 (1991), 59–96.
• ––––, Characters, dual pairs, and unitary representations, Duke Math. J. 69 (1993), 547–592.
• \lccS. Rallis and G. Schiffmann, Discrete spectrum of the Weil representation, Bull. Amer. Math. Soc. 83 (1977), 267–270.
• \lccD. A. Vogan Jr., “Associated varieties and unipotent representations” in Harmonic Analysis on Reductive Groups (Brunswick, Maine, 1989), Progr. Math. 101, Birkhäuser, Boston, 1991, 315–388.
• ––––, “The method of coadjoint orbits for real reductive groups” in Representation Theory of Lie Groups (Park City, Utah), IAS/Park City Math. Ser. 8, Amer. Math. Soc., Providence, 2000, 179–238.
• \lccN. R. Wallach, The analytic continuation of the discrete series, I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17; 19–37.
• \lccH. Weyl, The Classical Groups: Their Invariants and Representations, Princeton Univ. Press, Princeton, 1939.
• \lccH. Yamashita, “Cayley transform and generalized Whittaker models for irreducible highest weight modules” in Nilpotent Orbits, Associated Cycles and Whittaker Models for Highest Weight Representations, Astérisque 273, Soc. Math. France, Montrouge, 2001, 81–137.
• \lccC.-B. Zhu, Invariant distributions of classical groups, Duke Math. J. 65 (1992), 85–119.
• \lccC.-B. Zhu and J.-S. Huang, On certain small representations of indefinite orthogonal groups, Represent. Theory 1 (1997), 190–206.