ABSTRACT
This chapter explains the study of the extension of the representations from a subgroup H to the whole group G. It defines H to be observable in G when "every finite dimensional rational H-module can be embedded as an H-submodule in a rational G-module". The chapter describes the possibility of finding, inside an arbitrary H-stable ideal of k[G], a nonzero H-fixed element. It also defines observable subgroups and present different characterizations of observability in terms of ideals, representations and characters. The chapter describes the observability in terms of the surjectivity of the evaluation map associated to the induction functor, and establish some of its basic properties, e.g., the transitivity of observability along towers of subgroups. It considers two interrelated conditions that strengthen observability: split and strong observability. The chapter presents a proof of the usual geometric characterization of observability namely: H is observable in G if and only if the homogeneous space G/H is a quasi-affine variety.
