The SL(3, ℂ)-representation
variety R of a free group Fr arises naturally by considering surface group
representations for a surface with boundary. There is an SL(3, ℂ)-action on the
coordinate ring of R. The geometric points of the subring of invariants of this action
is an affine variety X. The points of X parametrize isomorphism classes of completely
reducible representations. The coordinate ring ℂ[X] is a complex Poisson algebra
with respect to a presentation of Fr imposed by the surface. In previous work, we
have worked out the bracket on all generators when the surface is a three-holed
sphere and when the surface is a one-holed torus. In this paper, we show how the
symplectic leaves corresponding to these two different Poisson structures
on X relate to each other. In particular, they are symplectically dual at a
generic point. Moreover, the topological gluing map that turns the three-holed
sphere into the one-holed torus induces a rank-preserving Poisson map on
ℂ[X].
