As we saw in Chapter 2, the study of simple KG-modules for a finite group G of Lie type (in the defining characteristic) can be reduced to the study of simple G-modules L(λ) with p-restricted highest weights. In the algebraic group setting, a number of ideas and techniques have been introduced in the hope of getting more complete information about the formal characters of these modules, based on the study of Weyl modules.

First we summarize the main results, most of which are developed systematically in [RAGS, II.6–11.8] (though in a different logical order):

• Weyl modules (3.1)

• Premet's comparison of weights of simple modules and Weyl modules when the highest weight is p-restricted (3.2)

• relationship with cohomology of line bundles on the flag variety (3.3)

• affine Weyl group (3.4–3.5)

• Linkage Principle and translation functors (3.6)

• Steinberg modules (3.7)

• contravariant form on a Weyl module (3.8)

• Jantzen filtrations and sum formula (3.9)

If p is not too small (say p ≥ h, the Coxeter number), Lusztig's 1979 Conjecture (3.11) offers the best hope for a uniform theoretical explanation of the characters in a setting essentially independent of p. At this writing it remains unproved in general. But it has been proved indirectly by Andersen–Jantzen–Soergel [18] for “sufficiently large” p (though without a definite estimate of how large p should be).

More direct computational methods, which are effective for very small p and low ranks, will be discussed in the next chapter.