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Let $W$ be an extended affine Weyl group. We prove that the minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some nice properties, generalizing results of Geck and Pfeiffer \cite{GP} 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 $T_{w_\co}$, where $\co$ 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 \cite{H99}.

Minimal length elements, affine Weyl groups, affine Hecke algebras