LEFT CELLS IN THE AFFINE WEYL GROUP Wa(D̃4)

The cells of affine Weyl groups have been studied for more than one decade. They have been described explicitly in cases of type Ãn ( n ≥ 1 ) [13][9] and of rank ≤ 3 [1][4][10]. But there are only some partial results for an arbitrary irreducible affine Weyl group [2][7][8][16][17]. In [18], we constructed an algorithm to find a representative set of left cells of certain crystallographic group W in a given two-sided cell. This provides us a practicable way to describe the cells of more groups. In the present paper, we shall apply it to the case when W is the affine Weyl group Wa(D̃4) ( or denoted by Wa for brevity ) of type D̃4. We shall give an explicit description for all the left cells of Wa by finding a representative set of left cells of Wa. Before this paper, Du Jie gave an explicit description for all the two-sided cells of Wa, but he was unable to find the left cells of this group [5]. Chen Chengdong recently described all the left cells of Wa in terms of certain special reduced expressions of elements [3]. Comparing with their results, our description on the cells of Wa is neater and easier expressable in nature. Moreover, by doing the above work, we develop some technical skill in performing the mentioned algorithm. In particular, we could avoid any computation of non-trivial Kazhdan-Lusztig polynomials throughout this work. The content of the present paper is organized as below. Section 1 is the preliminaries. Some basic concepts and results concerning our algorithm are stated there. In section 2, we introduce the alcove forms of elements of Wa and also state some properties of elements of Wa in terms of alcove forms, which are quite useful in the subsequent sections. Then in sections 3–5, we apply our algorithm to find a representative set Σ of left cells of Wa. Finally, in section 6, we describe all the left cells of Wa by making use of the set Σ.