Extended affine Weyl groups

In this paper we study the Weyl groups of reduced extended affine root syst ems, the root systems of extended affine Lie algebras. We start by describi ng the extended affine Weyl group as a semidirect product of a finite Wcyl group and a Heisenberg-like normal subgroup. This provides a unique express ion for the Weyl group elements (in terms of some naturally arisen transfor mations) which is crucial in the further study of extended affine Weyl grou ps. We use this to give a presentation, called a presentation by conjugatio n, for an important subclass of extended affine Weyl groups. Using a new no tion, called the index which is an invariant of the extended affine root sy stems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine roo t systems. (C) 1999 Academic Press.