Nahm's equations, configuration spaces and flag manifolds |
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Authors: | Michael Atiyah Roger Bielawski |
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Institution: | (1) Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, SCOTLAND E-mail: atiyah@maths.ed.ac.uk, GB;(2) Department of Mathematics, University of Glasgow, Glasgow G12, 8QW, SCOTLAND E-mail: R.Bielawski@maths.gla.ac.uk, GB |
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Abstract: | 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 S
U(2) to G. We then generalize this picture to include an arbitrary homomorphism from S
U(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.
Received: 8 February 2002 |
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Keywords: | : Berry-Robbins problem Hecke algebras Nahm's equations |
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