Nonreduced extended affine Weyl groups

Abstract

In this paper we study the structure of the Weyl groups of nonreduced extended affine root systems. We show that similar to the case of reduced types, an extended affine Weyl group W of type BCℓ is semidirect product of a finite Weyl group W (of type Bℓ ) and a Heisenberg-like normal subgroup H which is also a characteristic subgroup of W. Moreover, H is of the form H = HnH0, where both Hn and H0 are normal subgroups of H with Hn ∩ H0 ≠{1}, Hn is naturally isomorphic to the root lattice of a finite root system of type BCℓ. Furthermore, the semidirect product of W and Hn is isomorphic to the Weyl group of a Kac-Moody affine subroot system of R of type BCℓ. © 2003 Published by Elsevier Inc.