Invariant functions on Lie groups and Hamiltonian flows of surface group representations

In [7] it was shown that if n is the fundamental group ofa closed oriented surface S and G is Lie group satisfying very general conditions, then the space Hom(n, G)/G of conjugacy classes of representation n-+G has a natural symplectic structure. This symplectic structure generalizes the Weil-Petersson Kahler form on Teichmiiller space (taking G= PSL(2, lR)), the cup-product linear symplectic structure on H (S, lR) (when G= lR), and the Kahler forms on the Jacobi variety of a Riemann surface M homeomorphic to S (when G= U(l)) and the NarasimhanSeshadri moduli space of semistable vector bundles of rank n and degree 0 on M (when G= U(n)). The purpose of this paper is to investigate the geometry of this symplectic structure with the aid of a natural family of functions on Hom(n, G)/G. The inspiration for this paper is the recent work of Scott Wolpert on the WeilPetersson symplectic geometry of Teichmiiller space [18-20]. In particular he showed that the Fenchel-Nielsen "twist flows" on Teichmiiller space are Hamiltonian flows (with respect to the Weil-Petersson Kahler form) whose associated potential functions are the geodesic length functions. Moreover he found striking formulas which underscore an intimate relationship between the symplectic geometry of Teichmiiller space and the hyperbolic geometry (and hence the topology) of the surface. In particular the symplectic product of two twist vector fields (the Poisson bracket of two geodesic length functions) is interpreted in terms of the geometry of the surface. We will reprove these formulas of Wolpert in our more general context. Accordingly our proofs are simpler and not restricted to Teichmiiller space: while Wolpert's original proofs use much of the machinery of Teichmiiller space theory, we give topological proofs which involve the multiplicative properties of homology with local coefficients and elementary properties of invariant functions. Before stating the main results, it will be necessary to describe the ingredients of the symplectic geometry. Let G be a Lie group with Lie algebra g. The basic property we need concerning Gis the existence ofan orthogonal structure on G: an orthogonal structure on G is a nondegenerate symmetric bilinear form ~ : 9 x g-+ lR