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

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

Citations:

Zbl 0574.32032
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References:

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