Algebraic Homogeneous Spaces

Abstract

In this chapter, G denotes an arbitrary linear algebraic group (not supposed to be either connected nor reductive), and H⊆G a closed subgroup. We begin in §1 with the definition of an algebraic homogeneous space G/H as a geometric quotient, and prove its quasiprojectivity. We also prove some elementary facts on tangent vectors and G-equivariant automorphisms of G/H. In §2, we describe the structure of G-fibrations over G/H and compute Pic(G/H). Some related representation theory is discussed there: induction, multiplicities, the structure of $\Bbbk[G]$. Basic classes of homogeneous spaces are considered in §3. We prove that G/H is projective if and only if H is parabolic, and consider criteria of affinity of G/H. Quasiaffine G/H correspond to observable H, which may be defined by several equivalent conditions (see Theorem 3.12).