Transformation Groups

, Volume 4, Issue 4, pp 375–404 | Cite as

A convexity theorem for noncommutative gradient flows

  • C. Neidhardt
Article
Received:
Accepted:
  • 34 Downloads

Abstract

In this survey we shall prove a convexity theorem for gradient actions of reductive Lie groups on Riemannian symmetric spaces. After studying general properties of gradient maps, this proof is established by (1) an explicit calculation on the hyperbolic plane followed by a transfer of the results to general reductive Lie groups, (2) a reduction to a problem on abelian spaces using Kostant's Convexity Theorem, (3) an application of Fenchel's Convexity Theorem. In the final section the theorem is applied to gradient actions on other homogeneous spaces and we show, that Hilgert's Convexity Theorem for moment maps can be derived from the results.

Keywords

General Property Final Section Symmetric Space Homogeneous Space Topological Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AL92] D. Arnal and J. Ludwig,La convexité de l'application moment d'un groupe de Lie, J. Funct. Anal.105 (1992), 256–300.Google Scholar
  2. [At82] M. F. Atiyah,Convexity and commuting Hamiltonians, Bull. London Math. Soc.14 (1982), 1–15.Google Scholar
  3. [BBR92] A. M. Bloch, R. W. Brockett and T. S. Ratiu,Completely integrable gradient flows, Comm. Math. Phys.147/1 (1992), 57–74.Google Scholar
  4. [DKV83] J. J. Duistermaat, J. A. C. Kolk and V. S. Varadarajan,Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Comp. Math.49 (1983), 309–398.Google Scholar
  5. [FK94] J. Faraut and A. Koranyi,Analysis on Symmetric Cones, Oxford Mathematical Monographs, Oxford Science Publications, 1994.Google Scholar
  6. [GHL87] S. Gallot, D. Hulin and J. Lafontaine,Riemannian Geometry, Springer, Berlin, 1987.Google Scholar
  7. [Gr90] M. L. Gromov,Convex sets and Kähler manifolds, Advances in Differential Geometry, World Sci. Publ., 1990, pp. 1–38.Google Scholar
  8. [GS82] V. Guillemin and S. Sternberg,Convexity properties of the moment mapping, Invent. Math.67 (1982), 491–513.Google Scholar
  9. [HH96] P. Heinzner and A. T. Huckleberry,Kählerian potentials and convexity properties of the moment map, Invent. Math.126:1 (1996), 65–84.Google Scholar
  10. [Hel78] S. Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, London, 1978.Google Scholar
  11. [Hel84] S. Helgason,Groups and Geometric Analysis, Academic Press, London, 1984.Google Scholar
  12. [Hil94] J. Hilgert,A convexity theorem for boundaries of ordered symmetric spaces, Can. J. Math.46:4 (1994), 746–757.Google Scholar
  13. [HN91] J. Hilgert and K.-H. Neeb,Lie-Gruppen und Lie-Algebren, Vieweg Verlag, Wiesbaden, 1991.Google Scholar
  14. [HN95] J. Hilgert and K.-H. Neeb,Maximality of compression semigroups, Semigroup Forum50 (1995), 205–222.Google Scholar
  15. [HNP94] J. Hilgert, K.-H. Neeb and W. Plank,Symplectic convexity theorems and coadjoint orbits, Comp. Math.94 (1994), 129–180.Google Scholar
  16. [HO96] J. Hilgert and G. Olafsson,Causal Symmetric Spaces, Geometry and Harmonic Analysis, Academic Press, 1996.Google Scholar
  17. [Ki84] F. Kirwan,Convexity properties of the moment mapping III, Invent. Math.77 (1984), 547–552.Google Scholar
  18. [Ko73] B. Kostant,On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup.6 (1973), 413–455.Google Scholar
  19. [Ne93] K.-H. Neeb,Kähler geometry and convex sets, to appear, Proc. DMV-seminar “Kombinatorische Konvexgeometrie und torische Varietäten”, Blaubeuren 1993.Google Scholar
  20. [Ne94] K.-H. Neeb,Contraction semigroups and representations, For. Math.6 (1994), 233–270.Google Scholar
  21. [Ne95] K.-H. Neeb,On the convexity of the moment mapping for unitary highest weight representations, J. Funct. Anal.127 (1995), 301–325.Google Scholar
  22. [Ol90] G. Olafsson,Causal symmetric spaces, Preprint, Math. Gottingensis15 (1990).Google Scholar
  23. [Pa84] S. Paneitz,Determination of invariant convex cones in simple Lie algebras, Arkiv för Mat.21 (1984), 217–228.Google Scholar
  24. [Sch84] H. Schlichtkrull,Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Progress in Mathematics 49, Birkhäuser Verlag, 1984.Google Scholar
  25. [Wi89] N. J. Wildberger,Convexity and unitary representations of nilpotent groups, Invent. Math.98 (1989), 281–292.Google Scholar
  26. [War72] G. Warner,Harmonic Analysis on Semi-simple Lie Groups, Springer, New York, 1972.Google Scholar

Copyright information

© Birkhäuser Boston 1999

Authors and Affiliations

  • C. Neidhardt
    • 1
  1. 1.Kerpen-SindorfGermany

Personalised recommendations