Chapter IV. Homogeneous Spaces
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 2 definitions, 4 theorems, 2 lemmas, 3 constructions and 3 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Construction: actions on exterior powers
GL(W) and gl(W) act naturally on ∧^rW; stabilizing an r-plane M is equivalent to stabilizing the line ∧^rM.
Lemma: subspace stabilizers via exterior-power lines
For x∈GL(W) or gl(W), x stabilizes M iff its exterior-power action stabilizes the Plücker line ∧^rM.
Theorem: Chevalley stabilizer theorem
For closed H≤G there is a rational G-module V and a line L⊂V such that H is exactly the stabilizer of L and h is exactly its infinitesimal stabilizer.
Construction: passage from line stabilizers to projective orbits
A stabilizer line L defines a point [L]∈P(V); the orbit G·[L] is identified set-theoretically with G/H and is locally closed.
Definition: character group
A character is a morphism of algebraic groups χ:G→G_m; characters form the abelian group X(G) under multiplication.
Definition: semi-invariant and weight space
For a rational G-module V and χ∈X(G), V_χ={v | g·v=χ(g)v}; nonzero elements are semi-invariants of weight χ.
Lemma: independence of semi-invariant weight spaces
Distinct character weight spaces are linearly independent; only finitely many are nonzero in finite dimension.
Theorem: closed normal subgroups as kernels of rational representations
Every closed normal N◁G is the kernel of a rational representation ψ:G→GL(W), and n=Ker dψ.
Construction: homogeneous space G/H
For closed H≤G, embed H as a line stabilizer and take the orbit Y=G·[L]; the canonical map π:G→Y is surjective with fibres the left cosets of H.
Theorem: universal mapping property of G/H
The pair (π,Y) is universal for morphisms from G constant on H-cosets: each such morphism factors uniquely through Y.
Result: topology of the homogeneous-space quotient
The canonical quotient map π:G→G/H is open; the topology on G/H is the quotient topology for π.
Result: regular functions on a homogeneous space
Regular functions on open subsets of G/H are characterized by pullback to regular functions on G constant on H-cosets.
Results: complements and basic quotient geometry
The source records functoriality of homogeneous spaces, actions of G on G/H, and compatibility with subgroup inclusions/normal quotients.
Theorem: characteristic-zero Lie kernels and intersections
In characteristic zero, Lie(Kerφ)=Ker dφ for algebraic-group morphisms, and Lie(A∩B)=a∩b for closed subgroups.