zbMATH — the first resource for mathematics

Cartan decomposition of the moment map. (English) Zbl 1110.32008
The aim of the paper is to develop a geometric invariant theory for actions of real Lie groups on complex spaces. The authors accomplish this for certain real subgroups of a complex reductive group. The reductive group acts holomorphically on a complex space \(Z\). Assuming that \(Z\) admits an invariant Kähler structure and an equivariant moment mapping, they study the properties of an asociated moment mapping and consider its fibers and associated notions of semistable points. They also consider several topics pertaining to proper actions and compact isotropy groups. The results on proper actions are applied to obtain decompositions for groups and homogeneous spaces. The application relies on properties of distinguished strictly plurisubharmonic exhaustion.

32M05 Complex Lie groups, group actions on complex spaces
57S20 Noncompact Lie groups of transformations
Full Text: DOI
[1] Abels H. (1975). Parallelizability of proper actions, global K-slices and maximal compact subgroups. Math. Ann. 212:1–19 · Zbl 0287.57018 · doi:10.1007/BF01343976
[2] Azad, H., Loeb, J.J.: Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces. Indag. Math. N.S. 3–4, 365–375 (1992). · Zbl 0777.32008
[3] Azad H., Loeb J.J. (1993). Plurisubharmonic functions and the Kempf–Ness Theorem. Bull. Lond. Math. Soc. 25:162–168 · Zbl 0795.32002 · doi:10.1112/blms/25.2.162
[4] Azad H., Loeb J.J. (1999).Some applications of plurisubharmonic functions to orbits of real reductive groups. Indag. Math. N.S. 10: 473–482 · Zbl 0973.31007 · doi:10.1016/S0019-3577(00)87900-8
[5] Birkes D. (1971). Orbits of linear algebraic groups. Ann. Math. 93:459–475 · Zbl 0212.36402 · doi:10.2307/1970884
[6] Chevalley C. (1946). Theory of Lie Groups. Princeton University Press, Princeton · Zbl 0063.00842
[7] Guillemin V., Stenzel M. (1991). Grauert tubes and the homogeneous Monge-Ampère equation I. J. Diff. Geom. 34:561–570 · Zbl 0746.32005
[8] Heinzner P. (1991). Geometric invariant theory on Stein spaces. Math. Ann. 289:631–662 · Zbl 0728.32010 · doi:10.1007/BF01446594
[9] Heinzner P. (1993). Equivariant holomorphic extensions of real analytic manifolds. Bull. Soc. Math. France 121:445–463 · Zbl 0794.32022
[10] Heinzner P., Huckleberry A. (1996). Kählerian potentials and convexity properties of the moment map. Invent. Math. 126:65–84 · Zbl 0855.58025 · doi:10.1007/s002220050089
[11] Heinzner, P., Huckleberry, A.:Complex geometry of Hamiltonian actions (to appear) · Zbl 0855.58025
[12] Heinzner, P., Huckleberry, A., Kutzschebauch, F.: A real analytic version of Abels’ theorem and complexifications of proper Lie group actions. In: Complex Analysis and Geometry (Trento, 1993), Lecture Notes in Pure and Applied Mathematics 173, pp. 229–273 Dekker, New York (1996) · Zbl 0861.32011
[13] Heinzner P., Huckleberry A., Loose F. (1994). Kählerian extensions of the symplectic reduction. J. Reine und Angew. Math. 455:123–140 · Zbl 0803.53042
[14] Heinzner P., Loose F. (1994). Reduction of complex Hamiltonian G-spaces. Geom. Funct. Anal. 4:288–297 · Zbl 0816.53018 · doi:10.1007/BF01896243
[15] Heinzner P., Migliorini L., Polito M. (1998). Semistable quotients. Ann. Scuola Norm. Sup. Pisa 26:233–248 · Zbl 0922.32017
[16] Heinzner, P., Stötzel, H.: Semistable points with respect to real forms. (preprint, 2005) · Zbl 1129.32015
[17] Helgason S. (1978). Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York · Zbl 0451.53038
[18] Hochschild G. (1965). The structure of Lie groups. Holden-Day, San Francisco · Zbl 0131.02702
[19] Kempf, G., Ness, L.: The length of vectors in representation spaces. In: Lecture Notes in Math vol. 732, pp. 233–243 Springer, Berlin Heidelberg New York (1978) · Zbl 0407.22012
[20] Kirwan F. (1984). Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes vol. 31. Princeton University Press, Princeton · Zbl 0553.14020
[21] Lempert L., Szöke R. (1991). Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann. 290:689–712 · Zbl 0752.32008 · doi:10.1007/BF01459268
[22] Luna D. (1973). Slices étales. Bull. Soc. Math. France, Mémoire 33: 81–105 · Zbl 0286.14014
[23] Luna D. (1975). Sur certaines opérations différentiables des groupes de Lie. Am. J. Math. 97:172–181 · Zbl 0334.57022 · doi:10.2307/2373666
[24] Mostow G.D. (1955). Some new decomposition theorems for semisimple groups. Memoirs Am. Math. Soc. 14:31–54 · Zbl 0064.25901
[25] Mostow G.D. (1955). On covariant fiberings of Klein spaces. Am. J. Math. 77:247–278 · Zbl 0067.16003 · doi:10.2307/2372530
[26] Mumford D., Fogarty J., Kirwan F. (1994). Geometric invariant theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete 34. Springer, Berlin Heidelberg New York · Zbl 0797.14004
[27] Narasimhan R. (1962). The Levi problem for complex spaces, II. Math. Ann. 146:195–216 · Zbl 0131.30801 · doi:10.1007/BF01470950
[28] O’Shea L., Sjamaar R. (2000). Moment maps and Riemannian symmetric pairs. Math. Ann. 317:415–457 · Zbl 0985.37056 · doi:10.1007/PL00004408
[29] Palais R.S. (1961). On the existence of slices for actions of non-compact Lie groups. Ann. Math. 73:295–323 · Zbl 0103.01802 · doi:10.2307/1970335
[30] Richardson R.W., Slodowy P. (1990). Minimum vectors for real reductive algebraic groups. J. Lond. Math. Soc. 42:409–429 · Zbl 0675.14020 · doi:10.1112/jlms/s2-42.3.409
[31] Schwarz, G.W.: The topology of algebraic quotients. In: Kraft, H. et al. (eds.) Topological methods in algebraic transformation groups. Progress in Mathematics vol. 80, pp. 135–152 Birkhäuser Verlag, Basel-Boston (1989)
[32] Sjamaar R. (1995). Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. Math. 141(2):87–129 · Zbl 0827.32030 · doi:10.2307/2118628
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.