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

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- Trans. Amer. Math. Soc.
**368**(2016), 621-646 Request permission

## 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.## References

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## Additional Information

**Sean Rostami**- Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706-1325
- Email: srostami@math.wisc.edu, sean.rostami@gmail.com
- Received by editor(s): August 16, 2013
- Received by editor(s) in revised form: November 20, 2013, and November 29, 2013
- Published electronically: May 29, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 621-646 - MSC (2010): Primary 20F55; Secondary 20C08, 22E50
- DOI: https://doi.org/10.1090/tran/6342
- MathSciNet review: 3413877