## Characters of irreducible unitary representations of $\operatorname {U}(n, n+1)$ via double lifting from $\operatorname {U}(1)$

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- by Allan Merino
- Represent. Theory
**26**(2022), 325-369 - DOI: https://doi.org/10.1090/ert/597
- Published electronically: March 23, 2022
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## Abstract:

In this paper, we obtained character formulas of irreducible unitary representations of $U(n, n+1)$ by using Howe’s correspondence and the Cauchy–Harish-Chandra integral. The representations of $U(n, n+1)$ we are dealing with are obtained from a double lifting of a representation of $U(1)$ via the dual pairs $(U(1), U(1, 1))$ and $(U(1, 1), U(n, n+1))$.## References

- Anne-Marie Aubert and Tomasz Przebinda,
*A reverse engineering approach to the Weil representation*, Cent. Eur. J. Math.**12**(2014), no. 10, 1500–1585. MR**3224014**, DOI 10.2478/s11533-014-0428-8 - Florent Bernon and Tomasz Przebinda,
*Normalization of the Cauchy Harish-Chandra integral*, J. Lie Theory**21**(2011), no. 3, 615–702. MR**2858079** - Florent Bernon and Tomasz Przebinda,
*The Cauchy Harish-Chandra integral and the invariant eigendistributions*, Int. Math. Res. Not. IMRN**14**(2014), 3818–3862. MR**3239090**, DOI 10.1093/imrn/rnt058 - Abderrazak Bouaziz,
*Intégrales orbitales sur les groupes de Lie réductifs*, Ann. Sci. École Norm. Sup. (4)**27**(1994), no. 5, 573–609 (French, with English summary). MR**1296557**, DOI 10.24033/asens.1701 - 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** - Andrzej Daszkiewicz, Witold Kraśkiewicz, and Tomasz Przebinda,
*Nilpotent orbits and complex dual pairs*, J. Algebra**190**(1997), no. 2, 518–539. MR**1441961**, DOI 10.1006/jabr.1996.6910 - Thomas Enright, Roger Howe, and Nolan Wallach,
*A classification of unitary highest weight modules*, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 97–143. MR**733809** - Thomas J. Enright,
*Analogues of Kostant’s ${\mathfrak {u}}$-cohomology formulas for unitary highest weight modules*, J. Reine Angew. Math.**392**(1988), 27–36. MR**965055**, DOI 10.1515/crll.1988.392.27 - Harish-Chandra,
*Representations of semisimple Lie groups. III. Characters*, Proc. Nat. Acad. Sci. U.S.A.**37**(1951), 366–369. MR**42423**, DOI 10.1073/pnas.37.6.366 - Harish-Chandra,
*Representations of a semisimple Lie group on a Banach space. I*, Trans. Amer. Math. Soc.**75**(1953), 185–243. MR**56610**, DOI 10.1090/S0002-9947-1953-0056610-2 - Harish-Chandra,
*Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions*, Acta Math.**113**(1965), 241–318. MR**219665**, DOI 10.1007/BF02391779 - Harish-Chandra,
*Invariant eigendistributions on a semisimple Lie group*, Trans. Amer. Math. Soc.**119**(1965), 457–508. MR**180631**, DOI 10.1090/S0002-9947-1965-0180631-0 - Henryk Hecht,
*The characters of some representations of Harish-Chandra*, Math. Ann.**219**(1976), no. 3, 213–226. MR**427542**, DOI 10.1007/BF01354284 - Sigurdur Helgason,
*Differential geometry, Lie groups, and symmetric spaces*, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR**1834454**, DOI 10.1090/gsm/034 - Takeshi Hirai,
*The Plancherel formula for $\textrm {SU}(p,\,q)$*, J. Math. Soc. Japan**22**(1970), 134–179. MR**268331**, DOI 10.2969/jmsj/02220134 - Lars Hörmander,
*The analysis of linear partial differential operators. I*, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR**1996773**, DOI 10.1007/978-3-642-61497-2 - Roger Howe,
*Preliminaries i*, Unpublished. - Roger Howe,
*Wave front sets of representations of Lie groups*, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Springer-Verlag, Berlin-New York, 1981, pp. 117–140. MR**633659** - 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 - 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 - Anthony W. Knapp,
*Representation theory of semisimple groups*, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. An overview based on examples; Reprint of the 1986 original. MR**1880691** - Anthony W. Knapp,
*Lie groups beyond an introduction*, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR**1920389** - Stephen S. Kudla,
*On the local theta-correspondence*, Invent. Math.**83**(1986), no. 2, 229–255. MR**818351**, DOI 10.1007/BF01388961 - Jian-Shu Li,
*Singular unitary representations of classical groups*, Invent. Math.**97**(1989), no. 2, 237–255. MR**1001840**, DOI 10.1007/BF01389041 - Hung Yean Loke and Jiajun Ma,
*Invariants and $K$-spectrums of local theta lifts*, Compos. Math.**151**(2015), no. 1, 179–206. MR**3305312**, DOI 10.1112/S0010437X14007520 - Allan Merino,
*Characters of some unitary highest weight representations via the theta correspondence*, J. Funct. Anal.**279**(2020), no. 8, 108698, 70. MR**4124852**, DOI 10.1016/j.jfa.2020.108698 - Allan Merino,
*Transfer of characters in the theta correspondence with one compact member*, J. Lie Theory**30**(2020), no. 4, 997–1026. MR**4131128** - Allan Merino,
*Transfer of characters for discrete series representations of the unitary groups in the equal rank case via the Cauchy-Harish-Chandra integral,*arXiv:2101.02063, 2021. - Annegret Paul,
*Howe correspondence for real unitary groups*, J. Funct. Anal.**159**(1998), no. 2, 384–431. MR**1658091**, DOI 10.1006/jfan.1998.3330 - Annegret Paul,
*First occurrence for the dual pairs $(\textrm {U}(p,q),\textrm {U}(r,s))$*, Canad. J. Math.**51**(1999), no. 3, 636–657. MR**1701329**, DOI 10.4153/CJM-1999-029-6 - Tomasz Przebinda,
*Characters, dual pairs, and unipotent representations*, J. Funct. Anal.**98**(1991), no. 1, 59–96. MR**1111194**, DOI 10.1016/0022-1236(91)90091-I - 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 - Tomasz Przebinda,
*A Cauchy Harish-Chandra integral, for a real reductive dual pair*, Invent. Math.**141**(2000), no. 2, 299–363. MR**1775216**, DOI 10.1007/s002220000070 - Tomasz Przebinda,
*The character and the wave front set correspondence in the stable range*, J. Funct. Anal.**274**(2018), no. 5, 1284–1305. MR**3778675**, DOI 10.1016/j.jfa.2018.01.002 - Wilfried Schmid,
*On the characters of the discrete series. The Hermitian symmetric case*, Invent. Math.**30**(1975), no. 1, 47–144. MR**396854**, DOI 10.1007/BF01389847 - Binyong Sun and Chen-Bo Zhu,
*Conservation relations for local theta correspondence*, J. Amer. Math. Soc.**28**(2015), no. 4, 939–983. MR**3369906**, DOI 10.1090/S0894-0347-2014-00817-1 - David A. Vogan Jr.,
*Representations of real reductive Lie groups*, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR**632407** - Nolan R. Wallach,
*Real reductive groups. I*, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR**929683**

## Bibliographic Information

**Allan Merino**- Affiliation: Department of Mathematics, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 119077
- Address at time of publication: Department of Mathematics and Statistics, University of Ottawa, STEM Complex, 150 Louis-Pasteur Pvt., Ottawa, Ontario, K1N6N5, Canada
- MR Author ID: 1393076
- ORCID: 0000-0001-8545-9803
- Email: amerino@uottawa.ca
- Received by editor(s): March 17, 2021
- Received by editor(s) in revised form: September 14, 2021
- Published electronically: March 23, 2022
- Additional Notes: The author was supported by the MOE-NUS AcRF Tier 1 grants R-146-000-261-114 and R-146-000-302-114.
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 325-369 - MSC (2020): Primary 22E45; Secondary 22E46, 22E30
- DOI: https://doi.org/10.1090/ert/597
- MathSciNet review: 4398474