ABSTRACT
This chapter deals with the study of the geometric structure of a homogeneous space of the form G/H. It explains that a geometrically reductive subgroup of an affine algebraic group is observable. This is the first step in the proof that the quotient of an affine algebraic group by a geometrically reductive subgroup is affine. The chapter develops the case of a quasi-affine homogeneous spaces G/H the algebro- geometrical criterion for a quasi-affine variety to be affine by expressing it in terms of extensions of ideals from Hk[G] to k[G]. It relates the concept of exact subgroup with other concepts that classically have been important in invariant theory, namely integrals and Reynolds operators. The chapter provides the concept of exactness in the case of a unipotent group. The unipotency allows us to be more precise in relation to the exactness property.
