Chapter X. Structure of Reductive Groups
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 11 definitions, 13 theorems, 5 propositions, 7 lemmas, 7 corollaries, 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.
Definition: abstract root system and abstract Weyl group
A finite spanning subset Φ of a real vector space E, not containing 0, is a root system when only ±α are scalar multiples in Φ, each root has an associated reflection preserving Φ, and every reflection difference s_α(β)−β is an integral multiple of α. W(Φ) is generated by the root reflections.
Theorem: root system of a semisimple algebraic group
For semisimple G and E=R⊗_Z X(T), Φ(G,T) is an abstract root system of rank rank G, and its abstract Weyl group identifies with W(G,T).
Proposition: root subgroup shifts weights along root strings
For a rational representation ρ:G→GL(V) with kernel contained in Z(G), if V_λ is a T-weight space then ρ(U_α)V_λ lies in the sum of V_{λ+kα} over integral k.
Definition/result: base, simple roots and positive roots
A base Δ is a basis of the ambient root space such that every root has a unique integral expansion with coefficients all of one sign; its elements are simple roots and determine the positive roots. Bases exist and W acts simply transitively on them.
Comparison: bases, abstract chambers, cocharacter chambers and Borel subgroups
For fixed maximal T, bases of Φ, Weyl chambers in the root and cocharacter spaces, and Borel subgroups containing T correspond naturally and are each permuted simply transitively by W.
Theorem: generation by simple root rank-one subgroups
For a base Δ, G is generated by Z_α for α∈Δ, equivalently by T together with U_{±α} for simple roots α.
Definition: diagram-stabilizing automorphism subgroup
For semisimple G with fixed T⊆B, D≤Aut G consists of automorphisms stabilizing T and B; each induces an automorphism of the based root system, hence a Dynkin-diagram automorphism.
Theorem: automorphisms of a semisimple group modulo inner automorphisms
Aut G=(Int G)D, and the natural map from D to diagram automorphisms induces an injection Aut G/Int G into the finite diagram-automorphism group.
Theorem: minimal normal factors of a semisimple group
The minimal positive-dimensional closed connected normal subgroups G_i of semisimple G are finite in number, centralize one another, multiply onto G with finite kernel, and every closed connected normal subgroup is the product of the G_i it contains; moreover G=(G,G) and is generated by its root groups.
Corollary: simple factors and irreducible root-system components
The almost-direct decomposition of semisimple G into its minimal connected normal factors corresponds exactly to the decomposition of Φ into irreducible root systems.
Definition: simple/almost-simple algebraic group
A noncommutative connected algebraic group with no proper nontrivial closed connected normal subgroup is called simple, or almost simple when emphasizing that its abstract group may have a finite center.
Corollary/result: reductive group as center times derived group
A reductive G is the product Z(G)°·(G,G), with finite intersection; its semisimple derived part decomposes according to the irreducible components of Φ.
Definition: directly spanned by closed subgroups
H is directly spanned by H_1,…,H_r in the given order when the product morphism H_1×⋯×H_r→H is bijective.
Proposition: closed T-stable subgroups of a Borel unipotent radical
If H is a closed T-stable subgroup of U=R_u(B), then H is connected and is directly spanned, in any order, by exactly the root subgroups U_α whose root spaces lie in Lie(H).
Construction/result: root-sign factorization U=Uw Uw^+
For w∈W, the positive roots split according to the sign of w^{-1}α; the corresponding T-stable root-subgroup products give complementary directly spanning subgroups of U, with U equal to their product in either order.
Proposition: Bruhat decomposition in semisimple rank one
If G has semisimple rank one and s is the nontrivial Weyl element, then G=B ⊔ BsB. After fixing a representative n(s), every element of the nontrivial cell has a unique factorization u n(s)t u′ with the indicated rank-one unipotent factors and t∈T.
Theorem: Bruhat decomposition
For reductive G, G is the disjoint union ⨆_{w∈W}BwB, and two B-double cosets are equal iff their Weyl elements are equal.
Corollary: multiplication of a Bruhat cell by a simple reflection
For simple reflection s and w∈W, left multiplication of BwB by the rank-one cell BsB is contained in the union BwB∪BswB.
Corollary: intersection of two Borel subgroups
The intersection of any two Borel subgroups is connected and contains a maximal torus.
Theorem: unique Bruhat normal form
After choosing representatives n(w)∈N, every x∈G has a unique expression x=u′n(w)tu with w∈W, t∈T, u∈U and u′ in the root-sign subgroup U_w determined by w.
Proposition: big cell
Multiplication U^-×B→G is a bijection onto an open dense subset Ω, the big cell.
Result: big-cell product is an isomorphism of varieties
The product U^-×B→Ω is separable and birational and, using its equivariance, is an isomorphism of varieties.
Recorded result: ordered products of positive root groups
For any ordering α_1,…,α_r of the positive roots, multiplication U_{α_1}×⋯×U_{α_r}→U is an isomorphism of varieties.
Definition: Tits system (BN-pair)
A Tits system (G,B,N,S) has G generated by B,N; T=B∩N normal in N; W=N/T generated by involutions S; and axioms (T1) sBw⊆BwB∪BswB and (T2) sBs is not contained in B. W is its Weyl group and |S| its rank.
Definition: length and standard subsets in a Tits system
Length ℓ(w) is minimal word length in S; for I⊆S let W_I=⟨I⟩ and P_I=BW_IB.
Theorem: Bruhat decomposition for a Tits system
For every I⊆S, P_I is a subgroup; in particular P_S=G. The double cosets BwB are pairwise distinct.
Lemma: length change under a simple generator
For s∈S,w∈W, ℓ(sw)=ℓ(w)±1. The sign determines whether sBw lies in BswB alone or meets BwB, giving the basic double-coset length calculus.
Lemma: generation of PI from a reduced word
If a reduced expression for w uses simple generators in I, then P_I is generated by B together with wBw^{-1}, equivalently by B and representatives of those simple generators.
Lemma: intrinsic characterization of simple reflections
S is exactly the set of w for which B∪BwB is a subgroup; consequently S is a minimal generating set of W.
Lemma: conjugate-standard-parabolic containment
If wP_Iw^{-1}⊆P_J, then every representative of w in N lies in P_J.
Theorem: parabolic subgroups of a Tits system
The subgroups containing B are exactly P_I. Standard parabolics are conjugate only when equal, are self-normalizing, and inclusions P_I⊆P_J and W_I⊆W_J are equivalent to I⊆J.
Definition: Coxeter group presentation
A Coxeter group has generators s_i and defining relations (s_is_j)^{m(i,j)}=1, with m(i,i)=1 and symmetric m(i,j)≥2 or ∞.
Lemma: exchange condition
If w=s_1⋯s_r is reduced and ℓ(sw)<ℓ(w), then sw is obtained by deleting one generator from the reduced word.
Theorem: Weyl group of a Tits system is a Coxeter group
If m(i,j) is the order of s_is_j in W, the canonical epimorphism from the Coxeter group on S with these exponents to W is an isomorphism.
Lemma: normal subgroups and decompositions of the Coxeter generators
For a normal subgroup H◁G there is a partition S=I⊔J with I and J commuting elementwise and HB=P_I.
Definition: irreducible Weyl group of a Tits system
W is irreducible when S has no nontrivial partition into two elementwise commuting subsets.
Theorem: Tits simplicity criterion
If W is irreducible, G is generated by conjugates of a normal solvable U≤B, and G=(G,G), then G/Z is simple or trivial, where Z is the intersection of all conjugates of B.
Corollary: abstract simplicity modulo center of an almost-simple algebraic group
If G is simple/almost simple as an algebraic group and Z its finite center, then the abstract group G/Z is simple.
Definition: standard parabolic subgroup
Relative to B and base Δ, standard parabolics are the parabolic subgroups containing B, indexed by I⊆Δ as P_I=BW_IB.
Theorem: classification and roots of parabolic subgroups
Every parabolic is conjugate to a unique standard P_I; its roots relative to T are all positive roots together with the negative roots in the root subsystem spanned by I.
Definition: Levi decomposition and Levi factor
For P with V=R_u(P), a factorization P=L⋉V with L reductive is a Levi decomposition and L is a Levi factor.
Construction: canonical Levi factor of a standard parabolic
For standard P_I, if Z is the connected intersection of the kernels of roots in I, then L=C_G(Z) is reductive, has the root subsystem spanned by I, and P_I=L⋉R_u(P_I).
Theorem: existence and conjugacy of Levi factors
Every parabolic subgroup has a Levi decomposition, and any two Levi factors are conjugate by an element of its unipotent radical.
Construction: iterated unipotent-normalizer sequence P(U)
For closed unipotent U, repeatedly enlarge U_i by R_u(N_G(U_i)) and take the next normalizer until stabilization; the terminal normalizer is denoted P(U).
Lemma: connected solvable normalizer lies in a Borel containing the normalized subset
If A⊆B for a Borel B of H and connected solvable S≤H normalizes A, then S and A are contained in some common Borel subgroup of H.
Proposition: parabolicity criterion for a unipotent normalizer
If closed unipotent V lies in a Borel subgroup, N=N_G(V), and V contains R_u(N) with the stabilized identity-component condition of the construction, then N is parabolic and V=R_u(N).
Corollary: parabolic associated to a closed unipotent subgroup
If closed unipotent U lies in a Borel, then P(U) is parabolic, N_G(U)⊆P(U), and the terminal unipotent group is R_u(P(U)).
Corollary: self-unipotent-radical normalizer is parabolic
If connected unipotent U satisfies U=R_u(N_G(U)), then N_G(U) is parabolic.
Theorem: maximal subgroups and maximal unipotent subgroups of a reductive group
If H is a maximal proper closed subgroup of reductive G, either H° is reductive or H is parabolic. Every unipotent subgroup lies in a Borel subgroup; maximal unipotent subgroups are exactly Borel unipotent radicals.