Cells in affine Weyl groups and tilting modules

Abstract

Let $G$ be a reductive algebraic group over a field of positive characteristic. In this paper we explore the relations between the behaviour of tilting modules for $G$ and certain Kazhdan-Lusztig cells for the affine Weyl group associated with $G$. In the corresponding quantum case at a complex root of unity V. Ostrik has shown that the weight cells defined in terms of tilting modules coincide with right Kazhdan-Lusztig cells. Our method consists in comparing our modules for $G$ with quantized modules for which we can appeal to Ostrik's results. We show that the minimal Kazhdan-Lusztig cell breaks up into infinitely many "modular cells" which in turn are determined by bigger cells. At the opposite end we call attention to recent results by T. Rasmussen on tilting modules corresponding to the cell next to the maximal one. Our techniques also allow us to make comparisons with the mixed quantum case where the quantum parameter is a root of unity in a field of positive characteristic.