Minimal length elements of extended affine Coxeter groups, II

TL;DR

This paper extends the understanding of minimal length elements in extended affine Weyl groups, introduces class polynomials for affine Hecke algebras, and classifies conjugacy classes related to Lusztig's conjecture.

Contribution

It generalizes properties of minimal length elements to extended affine Weyl groups and introduces class polynomials for affine Hecke algebras.

Findings

Minimal length elements satisfy special properties in extended affine Weyl groups.

Class polynomials form a basis of the cocenter of affine Hecke algebra.

Classification of conjugacy classes related to Lusztig's conjecture.

Abstract

Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We then introduce the "class polynomials" for affine Hecke algebra $H$ and prove that $T_{w_\co}$, where $\co$ runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture \cite{L4}.