On the lowest two-sided cell in affine Weyl groups

Abstract

Bremke and Xi determined the lowest two-sided cell for affine Weyl groups with unequal parameters and showed that it consists of at most | W 0 | |W_{0}| left cells where W 0 W_{0} is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of parameters.