## Branching of metaplectic representation of $Sp(2, \mathbb R)$ under its principal $SL(2, \mathbb R)$-subgroup

HTML articles powered by AMS MathViewer

- by GenKai Zhang PDF
- Represent. Theory
**26**(2022), 498-514 Request permission

## Abstract:

We study the branching problem of the metaplectic representation of $Sp(2, \mathbb R)$ under its principle subgroup $SL(2, \mathbb R)$. We find the complete decomposition.## References

- Marc Burger, Alessandra Iozzi, and Anna Wienhard,
*Tight homomorphisms and Hermitian symmetric spaces*, Geom. Funct. Anal.**19**(2009), no. 3, 678–721. MR**2563767**, DOI 10.1007/s00039-009-0020-8 - 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** - Michel Duflo, Esther Galina, and Jorge A. Vargas,
*Square integrable representations of reductive Lie groups with admissible restriction to $\textrm {SL}_2(\Bbb R)$*, J. Lie Theory**27**(2017), no. 4, 1033–1056. MR**3646030** - Jacques Faraut and Adam Korányi,
*Analysis on symmetric cones*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR**1446489** - J. Faraut and A. Korányi,
*Function spaces and reproducing kernels on bounded symmetric domains*, J. Funct. Anal.**88**(1990), no. 1, 64–89. MR**1033914**, DOI 10.1016/0022-1236(90)90119-6 - Gerald B. Folland,
*Harmonic analysis in phase space*, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR**983366**, DOI 10.1515/9781400882427 - J. Frahm,
*Conformally invariant differential operators on Heisenberg groups and minimal representations*, Preprint, arXiv:2012.05952v1, 2020. - J. Frahm, C. Weiske, and G. Zhang,
*Principal series for Hermitian Lie groups induced from Heisenberg parabolic subgroups and intertwining operators*, in preparation. - Roger Howe and Eng-Chye Tan,
*Nonabelian harmonic analysis*, Universitext, Springer-Verlag, New York, 1992. Applications of $\textrm {SL}(2,\textbf {R})$. MR**1151617**, DOI 10.1007/978-1-4613-9200-2 - Roger Howe,
*On some results of Strichartz and Rallis and Schiffman*, J. Functional Analysis**32**(1979), no. 3, 297–303. MR**538856**, DOI 10.1016/0022-1236(79)90041-7 - 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 - Toshiyuki Kobayashi,
*Restrictions of unitary representations of real reductive groups*, Lie theory, Progr. Math., vol. 229, Birkhäuser Boston, Boston, MA, 2005, pp. 139–207. MR**2126642**, DOI 10.1007/0-8176-4430-X_{3} - Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw,
*Hypergeometric orthogonal polynomials and their $q$-analogues*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR**2656096**, DOI 10.1007/978-3-642-05014-5 - Bertram Kostant,
*The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group*, Amer. J. Math.**81**(1959), 973–1032. MR**114875**, DOI 10.2307/2372999 - Ottmar Loos,
*Jordan pairs*, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR**0444721**, DOI 10.1007/BFb0080843 - Bent Ørsted and Jorge A. Vargas,
*Branching problems for semisimple Lie groups and reproducing kernels*, C. R. Math. Acad. Sci. Paris**357**(2019), no. 9, 697–707 (English, with English and French summaries). MR**4018081**, DOI 10.1016/j.crma.2019.09.004 - Joe Repka,
*Tensor products of unitary representations of $\textrm {SL}_{2}(\textbf {R})$*, Amer. J. Math.**100**(1978), no. 4, 747–774. MR**509073**, DOI 10.2307/2373909 - Henrik Schlichtkrull,
*One-dimensional $K$-types in finite-dimensional representations of semisimple Lie groups: a generalization of Helgason’s theorem*, Math. Scand.**54**(1984), no. 2, 279–294. MR**757468**, DOI 10.7146/math.scand.a-12059 - Genkai Zhang,
*Branching coefficients of holomorphic representations and Segal-Bargmann transform*, J. Funct. Anal.**195**(2002), no. 2, 306–349. MR**1940358**, DOI 10.1006/jfan.2002.3957 - Genkai Zhang,
*Berezin transform on real bounded symmetric domains*, Trans. Amer. Math. Soc.**353**(2001), no. 9, 3769–3787. MR**1837258**, DOI 10.1090/S0002-9947-01-02832-X - Genkai Zhang,
*Tensor products of minimal holomorphic representations*, Represent. Theory**5**(2001), 164–190. MR**1835004**, DOI 10.1090/S1088-4165-01-00103-0 - Genkai Zhang,
*Principal series of Hermitian Lie groups induced from Heisenberg parabolic subgroups*, J. Funct. Anal.**282**(2022), no. 8, Paper No. 109399, 39. MR**4377991**, DOI 10.1016/j.jfa.2022.109399

## Additional Information

**GenKai Zhang**- Affiliation: Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, Göteborg University, SE-412 96 Göteborg, Sweden
- MR Author ID: 230134
- ORCID: 0000-0003-1147-3391
- Email: genkai@chalmers.se
- Received by editor(s): September 3, 2021
- Received by editor(s) in revised form: December 12, 2021, and February 7, 2022
- Published electronically: April 25, 2022
- Additional Notes: The research for this work was supported by the Swedish Science Council (VR)
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 498-514 - MSC (2020): Primary 22E45, 43A80, 43A90
- DOI: https://doi.org/10.1090/ert/609
- MathSciNet review: 4412276