Some translations in the second lowest two-sided cell of an affine Weyl group

We are interested in cell properties of translations in affine Weyl groups. We find out some translations in the second lowest two-sided cell of an affine Weyl group and use the translations to formulate a refinement of Jianyi Shi’s conjecture on the number of left cells in the second lowest two-sided cell. We verify the refinement for an affine Weyl group of type ${\tilde{A}}_{n-1}$ and type ${\tilde{G}}_2$.