Abstract
We address some conjectures and open problems in the ‘analysis of symmetries’ which include the study of non-commutative harmonic analysis and discontinuous groups for reductive homogeneous spaces beyond the classical framework:
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(1)
discrete series for non-symmetric homogeneous spaces G∕H;
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(2)
discontinuous groups Γ for G∕H beyond the Riemannian setting;
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(3)
analysis on pseudo-Riemannian locally homogeneous spaces.
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References
Y. Benoist and T. Kobayashi. Tempered homogeneous spaces IV. To appear in J. Inst. Math. Jussieu. Available also at arXiv: 2009.10391.
E. Calabi and L. Markus. Relativistic space forms. Ann. of Math.75, 63–76 (1962).
P. Delorme. Formule de Plancherel pour les espaces symétriques réductifs. Ann. of Math. (2)147, 417–452 (1998).
W.M. Goldman. Nonstandard Lorentz space forms. J. Differential Geom.21, 301–308 (1985).
B. Harris, Y. Oshima. On the asymptotic support of Plancherel measures for homogeneous spaces. arXiv:2201.11293.
K. Kannaka. Deformation of standard pseudo-Riemannian locally symmetric spaces. In preparation.
F. Kassel. Deformation of proper actions on reductive homogeneous spaces. Math. Ann.353, 599–632 (2012).
F. Kassel and T. Kobayashi. Poincaré series for non-Riemannian locally symmetric spaces. Adv. Math.287, 123–236 (2016).
F. Kassel and T. Kobayashi. Spectral analysis on standard locally homogeneous spaces. arXiv: 1912.12601.
T. Kobayashi. Proper action on a homogeneous space of reductive type. Math. Ann.285, 249–263 (1989).
T. Kobayashi. Singular unitary representations and discrete series for indefinite Stiefel manifoldsU(p, q; F)∕U(p − m, q; F). Mem. Amer. Math. Soc. 95, no. 462, vi+ 106 pp. (1992).
T. Kobayashi. Discrete decomposability of the restriction of \(A_{\mathfrak q}(\lambda )\) with respect to reductive subgroups and its applications. Invent. Math.117, 181–205 (1994).
T. Kobayashi. Discontinuous groups and Clifford–Klein forms of pseudo- Riemannian homogeneous manifolds. In: Algebraic and analytic methods in representation theory (Sønderborg, 1994), pp. 99–165, Perspect. Math., 17. Academic Press, San Diego, CA (1997).
T. Kobayashi. Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory. In: Translations, Series II, Selected Papers on Harmonic Analysis, Groups, and Invariants, 183, pp. 1–31, Amer. Math. Soc. (1998).
T. Kobayashi. Deformation of compact Clifford–Klein forms of indefinite-Riemannian homogeneous manifolds. Math. Ann.310, 395–409 (1998).
T. Kobayashi. Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups. J. Funct. Anal.152, 100–135 (1998).
T. Kobayashi. Discontinuous groups for non-Riemannian homogeneous spaces. In: Mathematics Unlimited–2001 and Beyond, pp. 723–747, Springer, Berlin (2001).
T. Kobayashi. Intrinsic sound of anti-de Sitter manifolds. Springer Proc. Math. Stat. 191, 83–99 (2016).
T. Kobayashi. Branching laws of unitary representations associated to minimal elliptic orbits for indefinite orthogonal group O(p, q). Adv. Math.388, Paper No. 107862, 38pp. (2021).
T. Kobayashi and T. Oshima. Finite multiplicity theorems for induction and restriction. Adv. Math.248, 921–944 (2013).
T. Kobayashi and T. Yoshino. Compact Clifford–Klein forms of symmetric spaces—revisited. Pure and Appl. Math. Quarterly1, 603–684, Special Issue: In Memory of Armand Borel (2005).
R.S. Kulkarni. Proper actions and pseudo-Riemannian space forms. Adv. Math.40, 10–51 (1981).
G. Margulis. Problems and conjectures in rigidity theory. In: Mathematics: frontiers and perspectives, pp. 161–174, Amer. Math. Soc. (2000).
T. Matsuki and T. Oshima. A description of discrete series for semisimple symmetric spaces. Adv. Stud. Pure Math.4, 331–390 (1984).
Y. Morita. Proof of Kobayashi’s rank conjecture on Clifford–Klein forms. J. Math. Soc. Japan71, no. 4, 1153–1171 (2019).
T. Okuda. Classification of semisimple symmetric spaces with proper \(SL(2,{\mathbb {R}})\)-actions. J. Differential Geom.94, 301–342 (2013).
N. Tholozan. Volume and non-existence of compact Clifford–Klein forms. arXiv:1511.09448v2, preprint.
K. Tojo. Classification of irreducible symmetric spaces which admit standard compact Clifford–Klein forms. Proc. Japan Acad. Ser. A Math. Sci.95, 11–15 (2019).
P.E. Trapa. Annihilators and associated varieties of \(A_{\mathfrak {q}}(\lambda )\) modules for U(p, q). Compositio Math.129, 1–45 (2001).
D.A. Vogan and G.J. Zuckerman. Unitary representations with nonzero cohomology. Compositio Math.53, 51–90 (1984).
N.R. Wallach. Real Reductive Groups. I. Pure Appl. Math. 132 Academic Press, Inc., Boston, MA (1988).
J.A. Wolf. Spaces of Constant Curvature. Sixth edition, xviii+424 pages, AMS Chelsea Publishing, Providence, RI (2011).
Acknowledgements
The author would like to express his sincere gratitude to his collaborators for the various projects mentioned in this article. This work was partially supported by Grant-in-Aid for Scientific Research (A) (18H03669), JSPS.
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Kobayashi, T. (2023). Conjectures on Reductive Homogeneous Spaces. In: Morel, JM., Teissier, B. (eds) Mathematics Going Forward . Lecture Notes in Mathematics, vol 2313. Springer, Cham. https://doi.org/10.1007/978-3-031-12244-6_17
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