Chapter IX. Centralizers of Tori
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 8 definitions, 4 theorems, 6 propositions, 3 lemmas, 8 corollaries, 4 constructions and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: regular and singular tori
For a torus S, let 𝔅^S be the Borel subgroups containing S. S is regular when 𝔅^S is finite and singular otherwise.
Definition: Weyl group relative to a torus
W(G,S)=N_G(S)/C_G(S), finite by torus rigidity. For maximal tori these groups are all isomorphic and are called the Weyl group of G.
Lemma: centralizer of a maximal torus lies in every containing Borel
If T is maximal and C=C_G(T), then C is contained in every Borel subgroup containing T.
Proposition: simple transitivity of Weyl group on Borel subgroups through T
For maximal T, W(G,T) acts simply transitively on 𝔅^T; hence |𝔅^T|=|W| and T is regular.
Proposition: Weyl groups and Borel sets under epimorphisms
An epimorphism G→G′ with T↦T′ induces surjections 𝔅T→𝔅{T′} and W(G,T)→W(G′,T′), bijective if the kernel lies in every Borel subgroup.
Construction: Weyl group action on characters, cocharacters and roots
N_G(T) acts on X(T),Y(T), descends to W, preserves the pairing, and permutes roots/root spaces by w·α(t)=α(n^{-1}tn), Ad n(g_α)=g_{wα}.
Proposition: regular torus criterion by solvability of its centralizer
A torus S is regular iff C_G(S) is solvable. Then C_G(S) lies in every Borel containing S, and for every maximal T⊇S one has 𝔅S=𝔅T.
Definition/construction: intersection subgroup I(T) and exterior roots
For maximal T, I(T) is the identity component of the intersection of all Borel subgroups containing T. Splitting g=Lie(I(T))⊕⊕_{α∈Ψ}g_α defines the finite set Ψ of root weights outside Lie(I(T)).
Definition: root-kernel torus
For α∈Ψ, T_α=(Ker α)° is a codimension-one subtorus of T.
Proposition: singular subtori detected by roots
For S⊆T, S is singular iff S is contained in T_α for some α∈Ψ; in particular the maximal singular subtori are the T_α.
Corollary: rank-one semisimple quotient of a root centralizer
For α∈Ψ and Z_α=C_G(T_α), the quotient Z_α/R(Z_α) is semisimple of rank one.
Definition: regular cocharacter
A one-parameter subgroup λ∈Y(T) is regular when λ(G_m) is a regular torus; Y(T)_reg denotes the regular cocharacters.
Proposition: root pairing criterion for regular cocharacters
λ∈Y(T) is regular iff ⟨α,λ⟩≠0 for every α∈Ψ.
Construction/result: extension of a cocharacter orbit in projective space
For a torus representation with weights χ_i and λ∈Y(T), the orbit map t↦λ(t)[v] extends to P¹; the 0 and ∞ limits are obtained by retaining the minimum and maximum λ-weight components and are λ(G_m)-fixed.
Result: cocharacters with separated weights have the same projective fixed points as T
Choosing λ so that distinct T-weights have distinct pairings makes the fixed points of λ(G_m) in P(V) coincide with the T-fixed points.
Lemma: hyperplane section of a projective variety
If irreducible projective X⊄P(W) with W codimension one and dim X>0, then X∩P(W) is nonempty and every irreducible component has codimension one.
Theorem: fixed points of a torus on an invariant projective variety
For torus T acting on irreducible closed X⊆P(V): if dim X≥1 the source obtains at least two T-fixed points; if dim X≥2 it obtains at least three.
Corollary: torus fixed points on G/P
For proper parabolic P and torus T, T fixes at least two points of G/P, and at least three when dim G/P>1; a maximal torus therefore lies in at least two/three corresponding Borel subgroups.
Corollary: lower bounds for the Weyl group from dim G/B
By simple transitivity, dim G/B=1 forces |W|=2, while higher-dimensional cases give the corresponding ≥3 lower bound recorded in the source.
Corollary: Borel subgroups through a maximal torus generate G
For maximal T, G is generated by all Borel subgroups containing T.
Definition: semisimple and reductive rank
rank_s G=rank(G/R(G)) and rank_red G=rank(G/R_u(G)).
Theorem: equivalent descriptions of semisimple rank one
For connected G and maximal T, rank_sG=1 iff |W|=2 iff |𝔅^T|=2 iff dim G/B=1 iff G/B≅P¹ iff there is an epimorphism G→PGL_2 with kernel identity component R(G).
Corollary: structure of reductive groups of semisimple rank one
If G is reductive of semisimple rank one, (G,G) is 3-dimensional semisimple, G=(G,G)Z(G) with finite intersection, C_G(T)=T, and any epimorphism to PGL_2 has kernel Z(G).
Construction: rank-one Weyl reflection attached to a singular torus
For α∈Ψ, Z_α=C_G(T_α) has Weyl group of order two; its nontrivial element s_α embeds canonically in W(G,T) and exchanges its two Borel subgroups through T.
Construction: Borel subgroup B(λ) from a regular cocharacter
A regular λ∈Y(T) selects a canonical T-fixed point on G/B by the ∞-limit construction, hence a Borel subgroup B(λ) containing T; the assignment is W-equivariant and onto 𝔅^T.
Definition: Weyl chamber
For B∈𝔅^T, C(B)={λ∈Y(T)_reg | B(λ)=B}; these nonempty sets partition the regular cocharacters and are permuted simply transitively by W.
Proposition: root sign detects the Borel in a rank-one centralizer
For α∈Ψ, the two Borels B_α,B′_α of Z_α through T can be labeled so that ⟨α,λ⟩>0 iff B(λ)∩Z_α=B_α.
Lemma: regular cocharacter separates nonproportional roots
For nonproportional α,β∈Ψ there is λ∈Y(T)_reg with ⟨α,λ⟩>0 and ⟨β,λ⟩<0.
Theorem: unipotent radical from intersection of Borels through T
If I(T) is the identity component of the intersection of B∈𝔅^T, then R_u(G) equals the unipotent part I(T)_u.
Corollary: torus centralizers in reductive groups
If G is reductive and S⊆T is a torus, C_G(S) is reductive. If S is regular then C_G(S)=T; hence Cartan subgroups are maximal tori and Z(G)⊆T.
Corollary: root-space, root-centralizer, center and root-lattice structure of a reductive group
For reductive G: Ψ=Φ(G,T) and Φ=−Φ; g=t⊕⊕{α∈Φ}g_α with dim g_α=1; the codimension-one singular tori are T_α; each Z_α=C_G(T_α) is reductive of semisimple rank one with Lie algebra t⊕g_α⊕g{−α}, and the Z_α generate G; Z(G)=∩_{α∈Φ}Ker α; and the root lattice in X(T) has rank rank_sG.
Corollary/definition: opposite Borel subgroup
For reductive G, every B∈𝔅^T has a unique opposite B^- with B∩B^-=T; writing B=TU and B-=TU- gives g=u^-⊕t⊕u.
Theorem/definition: root subgroups
For each α∈Φ there is a unique connected T-stable one-dimensional unipotent subgroup U_α with Lie algebra g_α; W-conjugation sends U_α to U_{wα}, and there is an admissible isomorphism x_α:G_a→U_α satisfying t x_α(c)t^{-1}=x_α(α(t)c). Together T and all U_α generate G.
Definition: admissible root-group parametrization
An isomorphism x_α:G_a→U_α satisfying the T-equivariance formula is called admissible; it is unique up to nonzero scalar.