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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras
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by Sean Rostami PDF
Trans. Amer. Math. Soc. 368 (2016), 621-646 Request permission


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.
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Additional Information
  • Sean Rostami
  • Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706-1325
  • Email:,
  • 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:
  • MathSciNet review: 3413877