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.


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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