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}.
Categories: math.RT