The Poisson Bracket of Length functions in the Hitchin Component

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Wolpert's cosine formula on Teichmüller space gives the Weil-Petersson Poisson bracket {lα, lβ} for geodesic length functions lα,lβ of closed curves α,β as the sum of the cosines of the angle of intersection of the associated geodesics. This was recently generalized to Hitchin representations by Labourie. In this paper, we give a short proof of this generalization using Goldman's formula for the Poisson bracket on representation varieties of surface groups into reductive Lie groups.