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 nqr = 1 2 |W0| should always hold. When Wa is either e An−1, n > 2, or of rank 6 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 nqr 6 12 |W0| in all the cases. §0. Introduction. 0.1. The two-sided cell Ωqr. Let W be a Coxeter group with S its distinguished generator set. In [7], Kazhdan and Lusztig introduced the concept of left, right and twosided cells in W in order to construct representations of W and the associated Hecke algebra H(W ). In [11], Lusztig further introduced a function a : W → N ∪ {∞} and proved that if W = Wa is an affine Weyl group, then the function a is constant on any two-sided cell of Wa and a(z) 6 ν for any z ∈ Wa, where ν is half the cardinal of the root system Φ associated to Wa. Let W(i) = {w ∈ Wa | a(w) = i} for any i > 0. It is known that the set W(ν) forms a single two-sided cell of Wa, which consists of |W0| left cells (see [20, Theorem 5.2], [21, Theorem 1.1]), where W0 is the Weyl group of Φ and |W0| is its cardinal. W(ν) is