ABSTRACT
The simplest platform to study the representation theory of a given algebraic object corresponds to the case where the representations are completely reducible. In the case of finite groups the complete reducibility of the representations is equivalent to the invertibility of the order of the group in the base field (Maschke's theorem) and it is also equivalent to the existence of a normalized integral for the corresponding group ring. This chapter presents the corresponding reducibility problem in the category of the rational representations of a given affine algebraic group G. It illustrates D. Hilbert's viewpoint with a brief sketch of one of his central ideas: the use of an averaging operator — whose existence is guaranteed by the complete reducibility of the representations — in order to produce invariants and to prove the finite generation of the ring of all invariants. In positive characteristic the complete reducibility of the representations was proved to be a very exceptional phenomenon.
