Characters of irreducible unitary representations of 𝑈(𝑛,𝑛+1) via double lifting from 𝑈(1)

Merino, Allan

doi:10.1090/ert/597

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