Left cells in affine Weyl groups

Abstract

We prove a property of left cells in certain crystallographic groups W , by which we formulate an algorithm to find a representative set of left cells of W in any given two-sided cell. As an illustration, we make some applications of this algorithm to the case where W is the affine Weyl group of type e F4. The cells of affine Weyl groups W , as defined by Kazhdan and Lusztig in [6], have been described explicitly in certain special cases: for W of rank 2, see [11]; for W of type An, see [16], [10]; for W of rank 3, see [1], [4]; for the cells with a-values 1, 2 and |Φ|/2 in a general W , see [2], [8], [9], [18], [19], where Φ is the root system determined by W . It is known that there exists a bijection between the set of two-sided cells in an affine Weyl group W and the set of unipotent classes in a certain complex reductive group G associated with W . It is also known that the value of the a-function on a two-sided cell of W is equal to the dimension of the variety of Borel subgroups of G containing an element of the corresponding unipotent class (see [14]). Thus for an affine Weyl group W , the two-sided cells of W are relatively well understood to certain extent. But the classification of left cells in a given two-sided cell of W is not known in general, even the number of these left cells. In the present paper, we shall introduce an algorithm to find a representative set of left cells of W ′ in a given two-sided cell, where W ′ is a group belonging to certain family of crystallographic