Mathematische Annalen

, Volume 342, Issue 2, pp 405–447 | Cite as

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

  • Florent SchaffhauserEmail author


The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkhäuser, Basel, 1995): given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom1(Σ = ∂M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations \({\mathcal{M}_{g,l}:=Hom_\mathcal{C}(\pi_{g,l}, U)/U}\)of the fundamental group π g,l of a punctured Riemann surface Σ g,l into an arbitrary compact connected Lie group U. This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involution\({\hat{\beta}}\) defined on \({\mathcal{M}_{g,l}}\) . We show that the involution \({\hat{\beta}}\) is induced by a form-reversing involution β defined on the quasi-Hamiltonian space \({(U\times U)^g \times\mathcal{C}_1\times\cdots\times \mathcal{C}_l}\) . The fact that \({\hat{\beta}}\) has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in Schaffhauser (A real convexity theorem for quasi-Hamiltonian actions, submitted, 25 p, 2007. The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.

Mathematics Subject Classification (2000)

53D30 53D12 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agnihotri S., Woodward C.: Eigenvalues of products of unitary matrices and quantum Schubert calculus. Math. Res. Lett. 5(6), 817–836 (1998)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Alekseev A., Kosmann-Schwarzbach Y., Meinrenken E.: Quasi-Poisson manifolds. Can. J. Math. 54(1), 3–29 (2002)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Alekseev A., Meinrenken E., Woodward C.: Linearization of Poisson actions and singular values of matrix products. Ann. Inst. Fourier (Grenoble) 51(6), 1691–1717 (2001)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Alekseev A., Meinrenken E., Woodward C.: Duistermaat-Heckman measures and moduli spaces of flat bundles over surfaces. Geom. Funct. Anal. 12(1), 1–31 (2002)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Alekseev A.Yu., Malkin A.Z.: Symplectic structures associated to Lie-Poisson groups. Commun. Math. Phys. 162(1), 147–173 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Alekseev A., Malkin A., Meinrenken E.: Lie group valued moment maps. J. Differ. Geom. 48(3), 445–495 (1998)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Atiyah M.F., Bott R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A 308(1505), 523–615 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bröcker T., Dieck T.: Representations of Compact Lie Groups. Graduate Texts in Mathematics, vol. 98. Springer, New York (1985)Google Scholar
  9. 9.
    Dostoglou S., Salamon D.A.: Self-dual instantons and holomorphic curves. Ann. of Math. (2) 139(3), 581–640 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Evens, S., Lu J.-H.: Thompson’s conjecture for real semisimple Lie groups. In: The breadth of Symplectic and Poisson geometry. Progr. Math., vol. 232, pp. 121–137. Birkhäuser, Boston (2005)Google Scholar
  11. 11.
    Falbel E., Wentworth R.A.: Eigenvalues of products of unitary matrices and Lagrangian involutions. Topology 45(1), 65–99 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Foth P.: A note on Lagrangian loci of quotients. Can. Math. Bull. 48(4), 561–575 (2005)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Goldman W.M.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2), 200–225 (1984)zbMATHCrossRefGoogle Scholar
  14. 14.
    Goldman W.M.: Geometric structures on manifolds and varieties of representations. In: Geometry of group representations (Boulder, CO, 1987). Contemp. Math., vol. 74, pp 169–198. American Mathematical Society, Providence (1988)Google Scholar
  15. 15.
    Guruprasad K., Huebschmann J., Jeffrey L., Weinstein A.: Group systems, groupoids, and moduli spaces of parabolic bundles. Duke Math. J. 89(2), 377–412 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ho N.-K.: The real locus of an involution map on the moduli space of flat connections on a Riemann surface. Int. Math. Res. Not. 61, 3263–3285 (2004)CrossRefGoogle Scholar
  17. 17.
    Ho N.-K., Melissa Liu C.-C.: Connected components of spaces of surface group representations. II. Int. Math. Res. Not. 16, 959–979 (2005)CrossRefGoogle Scholar
  18. 18.
    Klyachko A.A.: Stable bundles, representation theory and Hermitian operators. Selecta Math. (N.S.) 4(3), 419–445 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Klyachko, A.A.: Random walks on symmetric spaces and inequalities for matrix spectra. Linear Algebra Appl. 319(1–3), 37–59 (2000). Special Issue: Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem (Coimbra, 1999)Google Scholar
  20. 20.
    Lerman E., Sjamaar R.: Stratified symplectic spaces and reduction. Ann. Math. 134(2), 375–422 (1991)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Loos O.: Symmetric Spaces. II: Compact Spaces and Classification. W. A. Benjamin, Inc., New York (1969)Google Scholar
  22. 22.
    Meinrenken E., Woodward C.: Hamiltonian loop group actions and Verlinde factorization. J. Differ. Geom. 50(3), 417–469 (1998)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Narasimhan M.S., Seshadri C.S.: Holomorphic vector bundles on a compact Riemann surface. Math. Ann. 155, 69–80 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    O’Shea L., Sjamaar R.: Moment maps and Riemannian symmetric pairs. Math. Ann. 317(3), 415–457 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Salamon, D.: Lagrangian intersections, 3-manifolds with boundary, and the Atiyah-Floer conjecture. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994), pp. 526–536. Birkhäuser, Basel (1995)Google Scholar
  26. 26.
    Schaffhauser F.: Anti-symplectic involutions on quasi-Hamiltonian quotients. Trav. Math. 17, 57–64 (2007)Google Scholar
  27. 27.
    Schaffhauser, F.: Quasi-Hamiltonian quotients as disjoint unions of symplectic manifolds. In: Non-Commutative Geometry and Physics 2005— Proceedings of the International Sendai-Beijing JointWorkshop. World Scientific Press, New York (2007)Google Scholar
  28. 28.
    Schaffhauser, F.: A real convexity theorem for quasi-Hamiltonian actions, 25 p (2007, submitted).
  29. 29.
    Schaffhauser F.: Representations of the fundamental group of an l-punctured sphere generated by products of Lagrangian involutions. Can. J. Math. 59(4), 845–879 (2007)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Sleewaegen, P.: On generalized moment maps for symplectic compact group actions (preprint). Available at:
  31. 31.
    Will P.: The punctured torus and Lagrangian triangle groups in PU(2,1). J. Reine Angew. Math. 602, 95–121 (2007)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsKeio UniversityYokohamaJapan

Personalised recommendations