Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras

Abstract

Let W ~ = Λ ⋊ W ∘ \widetilde {W} = \Lambda \rtimes W_{\circ } be an Iwahori-Weyl group of a connected reductive group G G over a non-archimedean local field. The subgroup W ∘ W_{\circ } is a finite Weyl group, and the subgroup Λ \Lambda is a finitely generated abelian group (possibly containing torsion) which acts on a certain real affine space by translations. We prove that if w ∈ W ~ w \in \widetilde {W} and w ∉ Λ w \notin \Lambda , then one can apply to w w a sequence of conjugations by simple reflections, each of which is length-preserving, resulting in an element w ′ w^{\prime } for which there exists a simple reflection s s such that ℓ ( s w ′ ) , ℓ ( w ′ s ) > ℓ ( w ′ ) \ell ( s w^{\prime } ), \ell ( w^{\prime } s ) > \ell ( w^{\prime } ) and s w ′ s ≠ w ′ s w^{\prime } s \neq w^{\prime } . Even for affine Weyl groups, a special case of Iwahori-Weyl groups and also an important subclass of Coxeter groups, this is a new fact about conjugacy classes. Further, there are implications for Iwahori-Hecke algebras H \mathcal {H} of G G : one can use this fact to give dimension bounds on the “length-filtration” of the center Z ( H ) Z ( \mathcal {H} ) , which can in turn be used to prove that suitable linearly independent subsets of Z ( H ) Z ( \mathcal {H} ) are a basis.