Invariant functions on Lie groups and Hamiltonian flows of surface group representations. (English) Zbl 0619.58021
In a previous paper [Adv. Math. 54, 200–225 (1984; Zbl 0574.32032)], the author has shown that if \(\pi\) is the fundamental group of a closed oriented surface \(S\) and \(G\) is a Lie group satisfying very general conditions, then the space \(\operatorname{Hom}(\pi,G)/G\) of conjugacy classes of representations \(\pi\to G\) has a natural symplectic structure. (This structure generalizes the Weil-Petersson Kähler form on Teichmüller spaces, the Kähler form on Jacobi varieties of Riemann surfaces homeomorphic to \(S\) and other well-known symplectic structures.) The purpose of this paper is to investigate the geometry of this symplectic structure with the aid of a natural family of functions on \(\operatorname{Hom}(\pi,G)/G\).
Reviewer: Paul Godin (Bruxelles)
MSC:
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 57M05 | Fundamental group, presentations, free differential calculus |
| 43A99 | Abstract harmonic analysis |
| 22E99 | Lie groups |
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
Keywords:
invariant functions on Lie groups; Hamiltonian flows; surface group representations; Weil-Petersson Kähler form on Teichmüller spaces; Kähler form on Jacobi varietiesCitations:
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