Modular Representation Theory of Finite and p-Adic Groups

The following sections are included:

• CONTENTS
• FOREWORD
• PREFACE

We report on the main results about linear representations of finite reductive groups or finite groups of Lie type. Following the historical order, we comment on representations in the defining characteristic, ordinary characters and representations in non-defining characteristic.

In these notes, we give an overview of the representation theory of ρ-adic reductive groups with coefficients in fields of characteristic diffierent from ρ. We emphasize the case of GL(n) and its inner forms.

These notes are an introduction to the ρ-modular (or “mod-ρ”) repre-sentation theory of ρ-adic reductive groups. We will focus on the group GL₂(ℚρ), but we try to provide statements that generalize to an arbitrary ρ-adic reductive group G (for example, GL_n(ℚρ)).

This expository chapter is based on the lectures given at the program ‘Modular Representation Theory of Finite and ρ-Adic Groups’ at the National University of Singapore. We are concerned with recent results on representation theory and cohomology of KLR algebras, with emphasis on standard module theory.

The following sections are included:

• Introduction
• Acknowledgements
• Cyclotomic Hecke Algebras of Type A
• Cyclotomic Quiver Hecke Algebras of Type A
• Isomorphisms, Specht modules and categorification
• Seminormal bases and the KLR grading
• References