ABSTRACT
This chapter deals with the geometric structure of homogeneous spaces, i.e., varieties of the form G/H, where G is an affine algebraic group and H ⊂ G a closed subgroup. One of the main interests to study these objects is that they represent the orbits of the actions of affine groups on algebraic varieties. The chapter develops some preparatory material interrelating the representations of H with the representations of G; the basic results are due to C. It describes explicitly the manner in which H can be cut away from G by means of a finite number of semi-invariant polynomials with the same weight. The chapter shows that in the particular case that H is normal, the homogeneous space is in fact an affine algebraic group. The study of the maximal connected solvable subgroups of G — the Borel subgroups — has particular importance for the structure and representation theory of affine algebraic groups.
