The Two-Sided Cells of the Affine Weyl Group of Type Ãn

Abstract

Recently, J. Y. Shi [4] determined the left cells (in the sense of [1]) of the affine Weyl group of type Ãn; he has also obtained some partial results on the two-sided cells, which imply, in particular, that their number is at most equal to the number of partitions of n. In this paper we shall complete the results of Shi by proving the conjecture in [2,3.6] on the two-sided cells which implies in particular that the number of two-sided cells is exactly the number of partitions of n. The new ingredient in the proof is the function a(w) on an affine Weyl group which has been introduced in [3]. In the first three sections of this paper we shall recall the definition of a(w) and describe its connection with Gelfand-Kirillov dimension. In Section 4, we describe, following Vogan, an analogue of the notion of left cell, which we call left V-cell; in Section 5 we prove a finiteness theorem for the left V-cells of affine Weyl groups. Sections 6, 7 are concerned with affine Weyl groups of type Ãn.