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

Abstract:

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