## 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

Zbl 0574.32032
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### References:

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