On the Semisimplicity of Low-Dimensional Representations of Semisimple Groups in Characteristic p

There is an unfortunate conflict of terminology in the theory of algebraic groups in characteristic p: unlike in characteristic zero, the representations of a semisimple group need not be semisimple. Thanks to Chevalley [15], over any algebraically closed field we can associate to each dominant weight μ a representation Vμ satisfying Weyl’s character formula, but it is no longer, in general, irreducible. Springer [13] has shown that in one sense representation theory for large p stabilizes to the usual characteristic zero theory: for a fixed root system, a fixed dominant weight μ, and p 0, Vμ is irreducible. Unfortunately, there exist representations of low dimension which are not of the form Vμ, no matter how large p is taken. A typical example is the absolute Frobenius representation of SL(2):