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Minimal length elements of extended affine Weyl groups

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  19 August 2014

Xuhua He  and
Sian Nie
Xuhua He
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong email maxhhe@ust.hk Current address: Department of Mathematics, University of Maryland, College Park, MD 20742, USA
Sian Nie
Affiliation:
Institute of Mathematics, Chinese Academy of Sciences, 100190 Beijing, China email niesian@amss.ac.cn Current address: Max Planck Institute for Mathematics, Vivatsgasse 753111 Bonn, Germany
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Abstract

Let $W$ be an extended affine Weyl group. We prove that the minimal length elements $w_{{\mathcal{O}}}$ of any conjugacy class ${\mathcal{O}}$ of $W$ satisfy some nice properties, generalizing results of Geck and Pfeiffer [On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), 79–94] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some $p$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra $H$. We prove that $T_{w_{{\mathcal{O}}}}$, where ${\mathcal{O}}$ ranges over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H,H]$. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne–Lusztig varieties He [Geometric and cohomological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), 367–404].

Keywords

minimal length elementsaffine Weyl groupsaffine Hecke algebras

MSC classification

Primary: 20E45: Conjugacy classes 20F55: Reflection and Coxeter groups
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 11 , November 2014 , pp. 1903 - 1927
Copyright
© The Author(s) 2014 

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