## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Representation Theory of Algebraic Groups and Quantum Groups, T. Shoji, M. Kashiwara, N. Kawanaka, G. Lusztig and K. Shinoda, eds. (Tokyo: Mathematical Society of Japan, 2004), 1 - 16

### 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.

#### Article information

**Dates**

Received: 19 January 2002

First available in Project Euclid:
3 January 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1546542379

**Digital Object Identifier**

doi:10.2969/aspm/04010001

**Mathematical Reviews number (MathSciNet)**

MR2074586

**Zentralblatt MATH identifier**

1078.20043

#### Citation

Andersen, Henning Haahr. Cells in affine Weyl groups and tilting modules. Representation Theory of Algebraic Groups and Quantum Groups, 1--16, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/04010001. https://projecteuclid.org/euclid.aspm/1546542379