Extended Affine Root Systems III (Elliptic Weyl Groups)

Abstract

We give a presentation of an elliptic Weyl group W ( R ) (=the Weyl group for an elliptic root system*) R ) in terms of the elliptic Dynkin diagram Γ ( R, G ) for the elliptic root system. The presentation is a generalization of a Coxeter system: the generators are in one to one correspondence with the vertices of the diagram and the relations consist of two groups : i) elliptic Coxeter relations attached to the diagram, and ii) a flniteness condition on the Coxeter transformation attached to the diagram. The group defined only by the elliptic Coxeter relations is isomorphic to the central extension W ( R , G ) of W ( R ) by an infinite cyclic group, called the hyperbolic extension of W ( R ). *) an elliptic root system=a 2-extended affine root system (see the introduction and the remark at its end).