LOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP Wa( Ãn−1)

Let Γ be any canonical left cell of the affine Weyl group Wa of type e An−1, n > 1. We describe the lower boundary hyperplanes for Γ, which answer two questions of Humphreys. Let Wa be an affine Weyl group with Φ the root system of the corresponding Weyl group. Fix a positive root system Φ of Φ, there is a bijection from Wa to the set of alcoves in the euclidean space E spanned by Φ. We identify the elements of Wa with the alcoves (also with the topological closure of the alcoves) of E. According to a result of Lusztig and Xi in [6], we know that the intersection of any two-sided cell of Wa with the dominant chamber of E is exactly a single left cell of Wa, called a canonical left cell. When Wa is of type Ãn−1, n > 1, there is a bijection φ from the set of two sided cells of Wa to the set of partitions of n (see 2.4-2.6 and [7]). Recently, J. E. Humphreys raised the following Questions ([2]): Let Wa be the affine Weyl group of type Ãn−1, n > 1. (1) Could one find the set B(L) of all the lower boundary hyperplanes for any canonical left cell L of Wa ? Supported by Nankai Univ., the 973 Project of MST of China, the NSF of China, the SF of the Univ. Doctoral Program of ME of China, the Shanghai Priority Academic Discipline, and the CST of Shanghai (No. 03JC14027) Typeset by AMS-TEX 1