The second lowest two-sided cell in an affine Weyl group

Let Wa be an irreducible affine Weyl group with W0 the associated Weyl group. The present paper is to study the second lowest two-sided cell Ωqr of Wa. Let nqr be the number of left cells of Wa in Ωqr. We conjecture that the equality should always hold. When Wa is either , n⩾2, or of rank ⩽4, this equality can be verified by the existing data (see 0.3). Then the main result of the paper is to prove the inequality in all cases.