Modular representations and reflection subgroups

Abstract

The Hecke category is at the heart of several fundamental questions in modular representation theory. We emphasise the role of the "philosophy of deformations" both as a conceptual and computational tool, and suggest possible connections to Lusztig's "philosophy of generations". On the geometric side one can understand deformations in terms of localisation in equivariant cohomology. Recently Treumann and Leslie-Lonergan have added Smith theory, which provides a useful tool when considering mod p coefficients. In this context, we make contact with some remarkable work of Hazi. Using recent work of Abe on Soergel bimodules, we are able to reprove and generalise some of Hazi's results. Our aim is to convince the reader that the work of Hazi and Leslie-Lonergan can usefully be viewed as some kind of localisation to "good" reflection subgroups. These are notes for my lectures at the 2019 Current Developments in Mathematics at Harvard. Contents 1. Introduction 1 2. Philosophy of deformations 11 3. Affine reflection groups and Kazhdan-Lusztig theory 19 4. Localisation: equivariant, hyperbolic, Smith 33 5. The many faces of the Hecke category 46 6. The Hecke category and its localisations 54 7. Acknowledgements 69 References 69