Cells in affine weyl groups and tensor categories

Abstract

In this paper we construct a tensor (or monoidal) category for any two-sided cell in a finite or affine Weyl group. The isomorphism classes of simple objects of this category correspond to the elements of the two-sided cell; the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold, which imitates geometrically the definition of the ringJgiven in G. Lusztig (J. Algebra109, 1987, 536-548). Similarly, we construct, for any left cell, a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse (a right cell). This contains as a special case the category of equivariant perverse sheaves on an affine grassmannian, which, as a consequence of G. Lusztig (Astérisque101-102(1983), 208-209) is a tensor category for convolution (without truncation), closely related to the tensor category of representations of the group dual, in the sense of Langlands, to the group which gives rise to the affine grassmannian. But our construction can be specialized in other ways, and it gives a (conjectural) realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the Langlands dual group. © 1997 Academic Press.