The Partial Order on Two-sided Cells of Certain Affine Weyl Groups

In their famous paper [6], Kazhdan and Lusztig introduced the concept of equivalence classes such as left cell, right cell and two-sided cell in a Coxeter group W . We inherit the notations 6 L , 6 R , 6 LR , ∼ L , ∼ R and ∼ LR in [6]. Thus w ∼ LR y (resp. w ∼ L y, resp. w ∼ R y) means that the elements w, y ∈ W are in the same two-sided cell (resp. left cell, resp. right cell) of W , etc. Concerning an affine Weyl group Wa, Lusztig showed that the set Cell(Wa) of two-sided cells of Wa is in a natural 1-1 correspondence with the set U(G) of unipotent classes in the corresponding algebraic group G [11]. We know that Cell(Wa) is a poset under the relation 6 LR . Also, U(G) is a poset under the relation: v ≤ u in U(G) ⇐⇒ u ⊂ v, where v is the closure of the conjugacy class v in the variety of unipotent elements of G. Under the Lusztig’s correspondence, the two-sided cell c = {1Wa} ⊂ Wa is associated to the regular unipotent class of G, and the lowest two-sided cell W(v) (see [11]) of Wa is associated to the trivial class {1G} ⊂ G. Thus it is natural to formulate the following conjecture which was suggested by Lusztig (See [8, Conjecture D]) .