Cartan-decomposition subgroups of 𝑆𝑂(2,𝑛)

Oh, Hee; Morris, Dave

doi:10.1090/S0002-9947-03-03428-7

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

Cartan-decomposition subgroups of $\operatorname{SO}(2,n)$

Authors: Hee Oh and Dave Witte Morris
Journal: Trans. Amer. Math. Soc. 356 (2004), 1-38
MSC (2000): Primary 22E46; Secondary 20G20, 22E15, 57S20
Posted: August 25, 2003
MathSciNet review: 2020022
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For $G = \operatorname{SL} (3,\mathord{\mathbb{R} })$ and $G = \operatorname{SO}(2,n)$ , we give explicit, practical conditions that determine whether or not a closed, connected subgroup of has the property that there exists a compact subset of with . To do this, we fix a Cartan decomposition of , and then carry out an approximate calculation of $(KHK) \cap A^+$ for each closed, connected subgroup of .

[Ben] Yves Benoist, Actions propres sur les espaces homogènes réductifs, Ann. of Math. (2) 144 (1996), no. 2, 315–347 (French, with English and French summaries). MR 1418901 (97j:22023), http://dx.doi.org/10.2307/2118594
[Bor] Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
[BT] Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 0207712 (34 #7527)
[Hoc] G. Hochschild, The structure of Lie groups, Holden-Day Inc., San Francisco, 1965. MR 0207883 (34 #7696)
[Hm1] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. Graduate Texts in Mathematics, Vol. 9. MR 0323842 (48 #2197)
[Hm2] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773 (53 #633)
[Jac] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793 (26 #1345)
[KPS] Julia F. Knight, Anand Pillay, and Charles Steinhorn, Definable sets in ordered structures. II, Trans. Amer. Math. Soc. 295 (1986), no. 2, 593–605. MR 833698 (88b:03050b), http://dx.doi.org/10.1090/S0002-9947-1986-0833698-1
[Kb1] Toshiyuki Kobayashi, On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geom. Phys. 12 (1993), no. 2, 133–144. MR 1231232 (94g:22025), http://dx.doi.org/10.1016/0393-0440(93)90011-3
[Kb2] Toshiyuki Kobayashi, Criterion for proper actions on homogeneous spaces of reductive groups, J. Lie Theory 6 (1996), no. 2, 147–163. MR 1424629 (98a:57048)
[Kb3] Toshiyuki Kobayashi, Discontinuous groups and Clifford-Klein forms of pseudo-Riemannian homogeneous manifolds, Algebraic and analytic methods in representation theory (Sønderborg, 1994), Perspect. Math., vol. 17, Academic Press, San Diego, CA, 1997, pp. 99–165. MR 1415843 (97g:53061), http://dx.doi.org/10.1016/B978-012625440-2/50004-5
[Kos] Bertram Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455 (1974). MR 0364552 (51 #806)
[Kul] Ravi S. Kulkarni, Proper actions and pseudo-Riemannian space forms, Adv. in Math. 40 (1981), no. 1, 10–51. MR 616159 (84b:53047), http://dx.doi.org/10.1016/0001-8708(81)90031-1
[OW1] Hee Oh and Dave Witte, New examples of compact Clifford-Klein forms of homogeneous spaces of 𝑆𝑂(2,𝑛), Internat. Math. Res. Notices 5 (2000), 235–251. MR 1747110 (2001c:53071), http://dx.doi.org/10.1155/S1073792800000143
[OW2] H. Oh and D. Witte, Compact Clifford-Klein forms of homogeneous spaces of $\operatorname{SO}(2,n)$ (preprint).
[PS] Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697 (88b:03050a), http://dx.doi.org/10.1090/S0002-9947-1986-0833697-X
[Pog] Detlev Poguntke, Dense Lie group homomorphisms, J. Algebra 169 (1994), no. 2, 625–647. MR 1297165 (95m:22003), http://dx.doi.org/10.1006/jabr.1994.1300
[Rag] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. MR 0507234 (58 #22394a)
[vdD] Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348 (99j:03001)
[Var] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308 (85e:22001)
[W1] A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094. MR 1398816 (98j:03052), http://dx.doi.org/10.1090/S0894-0347-96-00216-0
[W2] A. J. Wilkie, O-minimality, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 633–636 (electronic). MR 1648050 (2000b:03115)
[Wit] Dave Witte, Superrigidity of lattices in solvable Lie groups, Invent. Math. 122 (1995), no. 1, 147–193. MR 1354957 (96k:22024), http://dx.doi.org/10.1007/BF01231442
[Zim] Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417 (86j:22014)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|