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 = P-1 (sic) infinity be the projective l ine. Let alpha is an element of H-2(X, Z). The moduli space of G-monopoles of topological charge alpha is naturally identified with the space M-b(X, a lpha) of based maps from (C, infinity) to (X, B) of degree alpha. The modul i space of G-monopoles carries a natural hyperkahler structure, and hence a holomorphic symplectic structure. It was explicitly computed by R. Bielaws ki in case G = SLn. We propose a simple explicit formula for another natura l symplectic structure on M-b(X, alpha). It is tantalizingly similar to R. Bielawski's formula, but in general (rank > 1) the two structures do not co incide. Let P superset of B be a parabolic subgroup. The construction of th e Poisson structure on Mb(X, a) generalizes verbatim to the space of based maps M = M-b(G/P, beta). In most cases the corresponding map (TM)-M-* --> T M is not an isomorphism, i.e. M splits into nontrivial symplectic leaves. T hese leaves are explicilty described.