Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 4, 2019

An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

Julius Grüning and Ralf Köhl
From the journal Advances in Geometry

Abstract

By [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane 𝓗 ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(ℝ). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact split type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac–Moody symmetric space G/K for an algebraically simply connected two-spherical split Kac–Moody group G, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

MSC 2010: 22.70; 20G44

  1. Communicated by: R. Löwen

Acknowledgements

The authors express their gratitude to the Deutsche Forschungsgemeinschaft for funding the research leading to this article via the project KO4323/13-1. They also thank an anonymous referee and Max Horn for several helpful comments on a preliminary version of this article.

References

[1] P. Abramenko, B. Mühlherr, Présentations de certaines BN-paires jumelées comme sommes amalgamées. C. R. Acad. Sci. Paris Sér. IMath. 325 (1997), 701–706. MR1483702 Zbl 0934.2002410.1016/S0764-4442(97)80044-4Search in Google Scholar

[2] C. S. Ballantine, Products of positive definite matrices. I. Pacific J. Math. 23 (1967), 427–433. MR0219555 Zbl 0211.3530210.2140/pjm.1967.23.427Search in Google Scholar

[3] P.-E. Caprace, Primitive symmetric spaces. Bull. Belg. Math. Soc. Simon Stevin12 (2005), 321–328. MR2173695 Zbl 1110.2000210.36045/bbms/1126195337Search in Google Scholar

[4] P.-E. Caprace, On 2-spherical Kac-Moody groups and their central extensions. Forum Math. 19 (2007), 763–781. MR2350773 Zbl 1140.2002810.1515/FORUM.2007.031Search in Google Scholar

[5] W. Freyn, T. Hartnick, M. Horn, R. Köhl, Kac–Moody symmetric spaces. To appear in Münster J. Math., arXiv:1702.08426 [math.GR]Search in Google Scholar

[6] H. Glöckner, R. Gramlich, T. Hartnick, Final group topologies, Kac-Moody groups and Pontryagin duality. Israel J. Math. 177 (2010), 49–101. MR2684413 Zbl 1204.2201410.1007/s11856-010-0038-5Search in Google Scholar

[7] T. Hartnick, R. Köhl, A. Mars, On topological twin buildings and topological split Kac–Moody groups. Innov. Incidence Geom. 13 (2013), 1–71. MR3173010 Zbl 1295.5101710.2140/iig.2013.13.1Search in Google Scholar

[8] O. Loos, Symmetric spaces. I, II. Benjamin, New York 1969. MR0239005/MR0239006 Zbl 0175.48601Search in Google Scholar

[9] K.-H. Neeb, On the geometry of standard subspaces. In: Representation theory and harmonic analysis on symmetric spaces, volume 714 of Contemp. Math., 199–223, Amer. Math. Soc. 2018. MR3847251 Zbl 0696763710.1090/conm/714/14330Search in Google Scholar

[10] R. Steinberg, Lectures on Chevalley groups. Yale University, New Haven, Conn. 1968. MR0466335 Zbl 1196.22001Search in Google Scholar

[11] J. Tits, Buildings of spherical type and finite BN-pairs. Springer 1974. MR0470099 Zbl 0295.20047Search in Google Scholar

[12] C. A. Weibel, An introduction to homological algebra. Cambridge Univ. Press 1994. MR1269324 Zbl 0797.1800110.1017/CBO9781139644136Search in Google Scholar

Received: 2018-05-16
Revised: 2018-08-15
Published Online: 2019-07-04
Published in Print: 2020-10-27

© 2020 Walter de Gruyter GmbH, Berlin/Boston