Nahm’s equations, configuration spaces and flag manifolds

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We give a positive answer to the Berry-Robbins problem for any compact Lie group G, i.e. we show the existence of a smooth W-equivariant map from the space of regular triples in a Cartan subalgebra to the flag manifold G/T . This map is constructed via solutions to Nahm’s equations and it is compatible with the S O(3) action, where S O(3) acts on G/T via a regular homomorphism from SU(2) to G. We then generalize this picture to include an arbitrary homomorphism from SU(2) to G. This leads to an interesting geometrical picture which appears to be related to the Springer representation of the Weyl group and the work of Kazhdan and Lusztig on representations of Hecke algebras.