Abstract
Let W a be an irreducible affine Weyl group with W 0 the associated Weyl group. The present paper is to study the second lowest two-sided cell Ω qr of W a . Let n qr be the number of left cells of W a in Ω qr . We conjecture that the equality n qr = 1 2 | W 0 | should always hold. When W a is either A ˜ n − 1 , 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 n qr ⩽ 1 2 | W 0 | in all cases.
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