Minimal length elements of extended affine Coxeter groups, II

Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\text{co}}$ of any conjugacy class $\text{co}$ 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 $T_{w_{\text{co}}}$, where $\text{co}$ runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H\mathrm{/}[H,H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig’s conjecture \cite{L4}.