The second lowest two-sided cell in the affine Weyl group [formula omitted

Abstract

Let (Wa,Sa) be an irreducible affine Weyl group with W0 the associated Weyl group. In my previous paper [24], I proved that the number nqr of left cells of Wa in the second lowest two-sided cell Ωqr satisfies the inequality nqr⩽12|W0| and conjectured that the equality should actually hold. In the present paper, we verify the conjecture when Wa=B˜n; we prove that any left cell of B˜n in Ωqr is left-connected, verifying a conjecture of Lusztig in our case; we show that Ωqr consists of all the extensions of wJ in the set B˜n−W(ν), where W(ν) is the lowest two-sided cell of B˜n and J⊂Sa is such that wJ is the longest element in the subgroup WJ of B˜n of type Dn.