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Nahm’s equations, configuration spaces and flag manifolds. (English) Zbl 1022.22007
In [Proc. R. Soc. Lond., Ser. A 453, 1771-1790 (1997; Zbl 0892.46084)] M. V. Berry and J. M. Robbins asked the question whether there is, for each integer \(n \geq 2\), a continuous map from the configuration space \(C_n({\mathbb R}^3)\) of \(n\) ordered distinct points of \({\mathbb R}^3\) to the flag manifold \(U(n)/T^n\) that is compatible with the action of the symmetry group \(\Sigma_n\). The first author gave in [The geometry of classical particles, in: Surveys in Differential Geometry 7, 1-15 (2000)] a positive answer to this problem by constructing explicitly such maps. In this paper, the authors present a generalization of the above to compact Lie groups. More precisely, let \(G\) be a compact Lie group and \(T\) a maximal torus of \(G\). Then the Weyl group \(W = N(T)/T\) acts freely on the flag manifold \(G/T\) and on \({\mathfrak t}^3 \setminus \Delta\), where \({\mathfrak t}\) is the Lie algebra of \(T\), \({\mathfrak t}^3 = {\mathfrak t} \otimes {\mathbb R}^3\), and \(\Delta\) is the singular set of the action of \(W\) on \({\mathfrak t}^3\). In the special case \(G = U(n)\) one has \({\mathfrak t}^3 \setminus \Delta = C_n({\mathbb R}^3)\) and \(W = \Sigma_n\). The authors construct an explicit map \({\mathfrak t}^3 \setminus \Delta \to G/T\) that is compatible with the action of \(W\) (actually, with an action of the bigger group \(W \times SU(2)\)). The map naturally arises from a general discussion about the space of solutions to Nahm’s equations on the half-line \({\mathbb R}_+\) satisfying suitable boundary conditions. It is known that such solution spaces are intimately related to hyper-Kähler metrics, and the authors discuss certain aspects related to the geometry of their construction. It also appears to be related to work by D. Kazhdan and G. Lusztig [Invent. Math. 80, 209-231 (1985; Zbl 0613.22003) and 87, 153-215 (1987; Zbl 0613.22004)], on representations of the Hecke algebras associated to Weyl groups, and the authors present some speculations about the geometric significance of these algebras and their modules.

MSC:
22E30 Analysis on real and complex Lie groups
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
20C08 Hecke algebras and their representations
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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