## 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

### Citations:

Zbl 0892.46084; Zbl 0613.22003; Zbl 0613.22004
Full Text: