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 \(\operatorname{Hom}(\pi,G)/G\).


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


Zbl 0574.32032
Full Text: DOI EuDML


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