Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-18T10:50:17.651Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Lagrangian Loci of Quotients

Published online by Cambridge University Press:  20 November 2018

Philip Foth
Philip Foth*
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089 e-mail: foth@math.arizona.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group.

Keywords

53D20Quotientsinvolutionsreal formsLagrangian loci
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005 , pp. 561 - 575
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Adams, J. and Vogan, D. A. Jr., L-groups, projective representations, and the Langlands classification. Amer. J. Math. 114(1992), 45–138.Google Scholar
[2] Alekseev, A., Mainrenken, E., and Woodward, C., Linearization of Poisson actions and singular values of matrix products. Ann. Inst. Fourier, (Grenoble) 51(2001), 1691–1717.Google Scholar
[3] Foth, P. and Lozano, G., The geometry of polygons in ℝ5 and quaternions. Geom. Dedicata 105(2004), 209–229.Google Scholar
[4] Foth, P. and Lu, J.-H., A Poisson structure on compact symmetric spaces. Comm. Math. Phys. 251(2004), 557–566.Google Scholar
[5] Gantmakher, F. R., The Theory of Matrices. AMS Chelsea Publishing, Providence, RI, 1959, (1998, reprint of the 1959 translation).Google Scholar
[6] Goldin, R. F. and Holm, T. S., Real loci of symplectic reductions. ArXiv: math.SG/0209111.Google Scholar
[7] Helgason, S., Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics 80, Academic Press, New York, 1978.Google Scholar
[8] MacDonald, I. G., Algebraic structure of Lie groups. In: Representation Theory of Lie Groups. London Mathematical Society Lecture Note Series 34, Cambridge, Cambridge University Press, 1979, pp. 91–150.Google Scholar
[9] Mumford, D., Fogarty, J., and Kirwan, F., Geometric Invariant Theory Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag, Berlin, 1994.Google Scholar
[10] O’Shea, L. and Sjamaar, R., Moment maps and Riemannian symmetric pairs. Math. Ann. 317(2000), 415–457.Google Scholar
[11] Richardson, R. W. and Slodowy, P. J., Minimum vectors for real reductive algebraic groups. J. London Math. Soc. 42(1990), 409–429.Google Scholar
[12] Serre, J.-P., Galois Cohomology. Springer-Verlag, Berlin, 1997.Google Scholar
[13] Sjamaar, R. and Lerman, E., Stratified symplectic spaces and reduction. Ann. of Math. 134(1991), 375–422.Google Scholar
[14] Wolf, J. A., The action of a real semisimple Lie group on a complex flag manifold, I: Orbit structure and holomorphic arc components. Bull. Amer.Math. Soc. 75(1969), 1121–1237.Google Scholar
[15] Xu, P., Dirac submanifolds and Poisson involutions. Ann. Sci. école Norm. Sup. (4) 36(2003), 403–430,Google Scholar