Abstract
Let W be a finite reflection group of a real vector space V. If W is crystallographic, then the quotient space V*//W appears in several contexts in geometry: i) in Lie theory as the quotient space of a simple Lie algebra by the adjoint Lie group action [Ch1,2] and ii) in complex geometry as the base space of the universal unfolding of a simple singularity [Br1]. Having these backgrounds, V*//W carries some distinguished geometric properties and structures, which, fortunately and also amusingly, can be described only in terms of the reflection group regardless whether W is crystallographic or not. We recall two of them:
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1.
The complexified regular orbit space (V*//W) regC is a K(π, 1)-space (Brieskorn [Br3], Deligne [De]). In other words, π 1((V*//W) regC is an Artin group (i.e. a generalized braid group [B-S][De]) and the universal covering space of (V*//W) regC is contractible (cf. also [Sa]).
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2.
The quotient space V*//W carries a flat structure (Saito [S3][S6])1. This means roughly that the tangent bundle of V*//W carries a flat metric J together with some additional structures. Nowadays, a flat structure without a primitive form is also called a Frobenius manifold structure with gravitational descendent (Dubrovin [Du], Manin [Ma1,2]).
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© 2004 Friedr. Vieweg & Sohn Verlag/GWV Fachverlage GmbH, Wiesbaden
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Saito, K. (2004). Uniformization of the orbifold of a finite reflection group. In: Hertling, K., Marcolli, M. (eds) Frobenius Manifolds. Aspects of Mathematics, vol 36. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-80236-1_11
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DOI: https://doi.org/10.1007/978-3-322-80236-1_11
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