CANONICAL LEFT CELLS AND THE SHORTEST LENGTH ELEMENTS IN THE DOUBLE COSETS OF WEYL GROUPS

Abstract. Let G be the general linear group GL(n,C), W0 the Weyl group of G and W the extended affine Weyl group of G. Then, it is well-known that W is a union of the double cosets W0xW0 as x moves over the set of dominant weights of W. It is also known that each double coset W0xW0 contains a unique element mx of the shortest length. These shortest length elements belong to what are called the canonical left cells. However, it is still an open problem to find the canonical left cell containing a given mx. One of the main purposes of this paper is to introduce a new approach to attack this question. In particular, we will present a conjecture which explicitly describes the canonical left cells containing an element mx. We will show that our conjecture is true for some specific types of mx. 1