2 - Simple Modules

Summary

To study the representations over K of an arbitrary finite group G, one usually concentrates first on those which are realized within the group algebra KG. The main examples are simple modules and projective modules.

When G is a finite group of Lie type, there are two natural approaches to the study of simple KG-modules:

• Describe them intrinsically in the setting of groups with split BN-pairs.

• Describe them as restrictions of simple modules for the ambient algebraic group G.

While the second approach is less direct, it has yielded (so far) much more detailed information than the first approach and will therefore be our main focus here. We defer until Chapter 7 the more self-contained development due to Curtis and Richen, based on BN-pairs.

Even though it is possible to classify the simple KG-modules in a coherent way from the algebraic group viewpoint, we still do not know in most cases their dimensions or (Brauer) characters. Modulo knowledge of the formal characters of simple G-modules (still incomplete in most cases), which we call standard character data for G, it is often possible to derive further results about the category of finite dimensional KG-modules: projectives, extensions, etc. This is usually the approach we follow, motivated by Lusztig's Conjecture for G (see 3.11 below).

After a detailed review of simple modules for the algebraic groups, following [RAGS], we turn to the finite groups.