A note on a symplectic structure on the space of G-monopoles

Let G be a semisimple complex Lie group with a Borel subgroup B. Let X = G/B be the flag manifold of G. Let C = ℙ¹ ∋ ∞ be the projective line. Let α ∈ H₂(X, ℤ). The moduli space of G-monopoles of topological charge α is naturally identified with the space script M sign _b(X, α) of based maps from (C, ∞) to (X, B) of degree α. The moduli space of G-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. It was explicitly computed by R. Bielawski in case G = SL_n. We propose a simple explicit formula for another natural symplectic structure on M_b(X, α). It is tantalizingly similar to R. Bielawski's formula, but in general (rank > 1) the two structures do not coincide. Let P ⊃ B be a parabolic subgroup. The construction of the Poisson structure on M_b(X, α) generalizes verbatim to the space of based maps script M sign = script M sign_b(G/P, β). In most cases the corresponding map script T sign* script M sign → script T sign script M sign is not an isomorphism, i.e. script M sign splits into nontrivial symplectic leaves. These leaves are explicilty described.