Chapter VIII. Borel Subgroups
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 3 definitions, 6 theorems, 2 propositions, 3 lemmas, 14 corollaries, 1 example, 1 construction and 2 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Recorded toolkit: complete/projective variety facts used for Borel theory
The source recalls that closed subvarieties/products/images behave well under completeness, projective varieties are complete, homogeneous spaces are quasiprojective, and complete affine varieties are finite.
Lemma: completeness transfers across bijective equivariant homogeneous morphisms
If connected G acts transitively on irreducible X,Y and φ:X→Y is a bijective G-equivariant morphism with Y complete, then X is complete.
Theorem: Borel fixed point theorem
A connected solvable algebraic group acting on a nonempty complete variety has a fixed point.
Definition: Borel subgroup
A Borel subgroup is a maximal closed connected solvable subgroup of G; Borel subgroups of G and G° coincide.
Theorem: conjugacy of Borel subgroups and projectivity of G/B
For any Borel subgroup B of connected G, G/B is projective and every Borel subgroup is G-conjugate to B.
Corollary: maximal tori and maximal connected unipotent subgroups
The maximal tori and maximal connected unipotent subgroups of G are precisely those occurring in Borel subgroups; each class is conjugate in G.
Definition: rank of a connected algebraic group
rank G is the common dimension of maximal tori of G.
Definition: parabolic subgroup
A closed P≤G is parabolic when G/P is projective, equivalently complete.
Corollary: parabolic subgroups are exactly those containing a Borel subgroup
A closed subgroup P is parabolic iff it contains a Borel subgroup; connected H is Borel iff H is solvable and G/H is projective.
Example/construction: parabolics of GLn as partial-flag stabilizers
For GL_n with standard upper triangular B, stabilizers of standard partial flags are parabolic subgroups containing B.
Corollary: Borel/parabolic/torus/unipotent subgroups under epimorphisms
An epimorphism of connected algebraic groups maps Borel subgroups, parabolics, maximal tori and maximal connected unipotent subgroups onto subgroups of the same type, and every such target subgroup arises this way.
Proposition: automorphism fixing a Borel subgroup pointwise
An automorphism of connected G fixing every element of a Borel subgroup is the identity.
Corollary: center versus a Borel centralizer
Z(G)°⊆Z(B)⊆C_G(B)=Z(G); later equality Z(B)=Z(G) is obtained.
Proposition: nilpotent Borel criteria
If a Borel subgroup is nilpotent then it equals G; equivalently G is nilpotent iff it has a unique maximal torus.
Corollary/definition: connected centralizer of a maximal torus as Cartan subgroup
For maximal torus T, C_G(T)° is nilpotent and self-normalizing up to identity component; it is called a Cartan subgroup at this stage.
Lemma: density criterion from a homogeneous fixed-point set
For closed connected H≤G, the union of conjugates X=⋃_g gHg^{-1} is closed if G/H is complete; if an element of H fixes only finitely many points of G/H then X contains a dense open subset of G.
Theorem: density theorem for Borel, torus, unipotent and Cartan conjugates
For connected G, the union of conjugates of a Borel subgroup is all G; the union of conjugates of a maximal torus is the semisimple locus, that of the Borel unipotent radical is the unipotent locus, and conjugates of a Cartan subgroup contain a dense open subset.
Corollary: semisimple/unipotent elements lie in connected nilpotent groups
Every semisimple or unipotent element of connected G lies in a connected nilpotent subgroup.
Corollary: center of a Borel subgroup
For any Borel subgroup B of connected G, Z(B)=Z(G).
Theorem: connectedness of torus centralizers
For any torus S in connected G, C_G(S) is connected.
Corollary/definition: Cartan subgroups are centralizers of maximal tori
The Cartan subgroups of connected G are exactly C_G(T) for maximal tori T; all are conjugate.
Corollary: commuting semisimple element and torus lie in a torus
If semisimple x centralizes a torus S, then x and S are contained in a common torus.
Theorem: centralizer action on torus-fixed components of G/B
For torus S, C=C_G(S), and an irreducible component Y of the fixed-point set (G/B)^S, C acts transitively on Y.
Corollary: Borel subgroups of torus centralizers
If B contains S, then B∩C_G(S) is a Borel subgroup of C_G(S), and every Borel subgroup of C_G(S) arises in this way.
Result: Cartan subgroup structure summary
For maximal T, C_G(T)=T⋉C_u with unique maximal torus T; Cartan subgroups are conjugate, nilpotent, and their conjugates contain a dense open subset.
Lemma: identity component of Borel normalizer
For Borel B, N_G(B)°=B.
Theorem: normalizer theorem for Borel subgroups
Every Borel subgroup is self-normalizing: N_G(B)=B.
Corollary: Borel subgroups are maximal abstract solvable subgroups among containing groups
A Borel subgroup is maximal among solvable subgroups of G, without assuming competitors connected or closed.
Corollary: parabolic subgroups are connected and self-normalizing
If P is parabolic then P=N_G(P), hence P is connected.
Corollary: parabolics containing a common Borel are not nontrivially conjugate
If parabolics P,Q both contain B and are G-conjugate, then P=Q.
Corollary: Borel subgroup normalizes exactly its unipotent radical
For Borel B with U=R_u(B), N_G(U)=B.
Construction/result: variety of Borel subgroups
The set 𝔅 of all Borel subgroups is identified G-equivariantly with G/B by xBx^{-1}↔︎xB; Borel subgroups containing H correspond to H-fixed points on G/B.