Chapter XII. Survey of Rationality Properties and Appendix
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 24 definitions, 11 theorems, 2 lemmas, 1 example and 18 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: k-closed subset and Zariski k-topology
For k⊆K, a closed subset of affine space is k-closed when it is the zero set of polynomials with coefficients in k; these sets define the Zariski k-topology.
Definition: variety defined over k
An affine variety is defined over k when its radical vanishing ideal is generated by its intersection with k[T]; intrinsically this gives a k-form k[X] with K[X]≅K⊗_k k[X], and the notion extends by affine coverings.
Lemma: k-closed implies definition after finite purely inseparable extension
Every k-closed affine algebraic set is defined over a finite purely inseparable extension of k. Hence over perfect k, k-closed and defined over k coincide.
Definition: k-rational points
For a variety X defined over k, X(k) is its intrinsically defined set of k-rational points; in affine coordinates it is X∩k^n.
Definition: morphism defined over k
A morphism between varieties defined over k is a k-morphism when its affine coordinate maps are defined over k; this is intrinsic and local on affine coverings.
Definition/result: algebraic group defined over k
A k-group is an algebraic group whose group variety, multiplication and inversion are defined over k. G(k) is then an abstract subgroup; the standard linear groups are defined over their prime field.
Recorded result: identity component of a k-group
If G is a k-group then G° is defined over k.
Recorded result: image of a k-homomorphism
The image of a k-morphism of k-groups is a k-subgroup; the kernel need not be defined over k over arbitrary fields.
Recorded result: group generated by k-defined morphic images
In the generation proposition of §7.5, k-defined input varieties and k-morphisms generate a subgroup defined over k.
Definition/result: k-morphic action and rationality of transporters
An action is k-morphic when group, variety and action map are defined over k. For k-rational source subsets and k-closed target sets, transporters, corresponding centralizers and fixed sets are k-closed.
Recorded result: orbits of a k-group action
For a k-group acting k-morphically, the orbits are varieties defined over k and orbit maps are k-morphisms.
Recorded theorem: linearization over k
Finite-dimensional stable enlargements can be chosen over k, and every k-group admits a k-isomorphism onto a k-subgroup of some GL_n.
Recorded result: k-structure on the Lie algebra
If G is a k-group, Lie(G) has a natural k-form g(k) with g≅K⊗_k g(k).
Recorded result: adjoint representation over k
For a k-group G, Ad:G→GL(g) is a k-morphism relative to the natural k-structure on g.
Recorded theorem: Chevalley stabilizer representation over k
If G and H are defined over k, Chevalley’s stabilizer representation can be chosen with k-structure, k-defined stabilizer line and k-morphic action.
Recorded theorem: normal subgroup kernel representation over k
If G and closed normal N are defined over k, the representation with kernel N can be chosen as a k-morphism into a k-form of GL(W).
Recorded result: homogeneous quotient over k
For k-groups G,H with H closed, the homogeneous space G/H and canonical quotient map can be chosen defined over k.
Recorded result: Jordan decomposition over perfect fields
For a k-rational matrix or element of a k-group, semisimple/unipotent and semisimple/nilpotent Jordan parts are k-rational when k is perfect; over arbitrary k a purely inseparable extension may be required. The unipotent locus is k-closed.
Definition: k-torus, k-split and k-anisotropic torus
A torus defined over k is k-split when its k-defined characters span its coordinate algebra, equivalently it is k-isomorphic to G_m^d; it is k-anisotropic when it has no nontrivial k-defined character.
Theorem: splitting and split/anisotropic factors of a k-torus
Every k-torus splits over a finite Galois extension. It has unique largest k-split and largest connected k-anisotropic subtori T′,T″ with T=T′T″; T″ is the identity component of the intersection of kernels of all k-defined characters.
Definition: k-split and k-anisotropic reductive group
A reductive k-group is k-split when it has a k-split maximal torus and compatible root-subgroup parametrizations over k; it is k-anisotropic when it has no positive-dimensional k-split torus.
Theorem: basic rationality theorems for connected k-groups
A connected k-group has a maximal torus defined over k. A reductive k-group is k-split iff some maximal torus is k-split and therefore splits over a finite Galois extension. Centralizers of k-tori are reductive and k-defined and the torus lies in a k-defined maximal torus. Under the stated infinitude/perfectness hypotheses, G(k) is Zariski dense.
Definition: k-rank, k-roots and k-Weyl group
Choose a maximal k-split torus S. Its dimension is rank_k G. The nonzero weights of Ad S are the k-roots; with Z=C_G(S), N=N_G(S), the finite quotient W_k=N/Z is the k-Weyl group.
Definition/result: nonreduced root system and type BC
Dropping the axiom excluding 2α alongside α gives nonreduced root systems. The only irreducible nonreduced family is BC_r, with no new Weyl groups beyond types B_r/C_r.
Theorem: Borel–Tits theorem on maximal k-split tori and relative roots
All maximal k-split tori are conjugate under G(k); Z=C_G(S) is a reductive k-group with k-anisotropic derived group; the k-roots form a possibly nonreduced abstract root system with Weyl group W_k.
Theorem: minimal k-parabolics
Minimal parabolic k-subgroups are G(k)-conjugate. Each contains a maximal k-split torus S, with C_G(S) a Levi factor; its unipotent radical is k-defined and its k-roots form a positive subsystem.
Theorem: relative Bruhat decomposition and Tits system
For minimal k-parabolic P with maximal k-split S, U=R_u(P), N=N_G(S), Z=C_G(S), rational representatives realize W_k and G(k) has a relative Bruhat decomposition; P(k),N(k) determine a Tits system.
Recorded result: rational points of anisotropic groups over perfect fields
If k is perfect and G is k-anisotropic, every element of G(k) is semisimple.
Definition/recorded result: elementary subgroup generated by minimal-parabolic unipotent radicals
For non-anisotropic G, G* is generated by U(k) over unipotent radicals of minimal k-parabolics. The source records Tits’s simplicity result for G/Z(G) in the simple case for most k, and notes that G(k)=G* often but not always holds for simply connected semisimple groups.
Example: relative structure of special orthogonal groups
For G=SO(q), char K≠2, the Witt decomposition of q exhibits a maximal k-split torus of dimension equal to the Witt index, its centralizer as the split torus times an anisotropic orthogonal factor, and explicit minimal k-parabolics/unipotent radicals; split and quasisplit cases are read off from the anisotropic dimension.
Definition: k-quasisplit group
A semisimple k-group is k-quasisplit when it has a Borel subgroup defined over k.
Recorded classification: possible nonsplit quasisplit types and relative root systems
Nonsplit quasisplit simple groups occur only when the absolute Dynkin diagram has nontrivial automorphisms. The source records the foldings A→C/BC, D→B (and D_4→G_2), and E_6→F_4, together with existence of such forms.
Recorded result: twisted finite groups of Lie type
Over finite fields the nonsplit quasisplit forms yield the twisted finite groups of Lie type; the source also records the Ree–Suzuki exceptional twists in characteristics 2 and 3.
Theorem/recorded result: Lang theorem
For a connected algebraic group over finite k of cardinality q, the Lang map x↦x^{-1}F_q(x) is surjective.
Theorem: semisimple groups over finite fields are quasisplit
If k is finite, every semisimple k-group is k-quasisplit.
Theorem: real points of algebraic groups
For closed G≤GL_n(C) defined over R, G(R) is a closed real Lie subgroup; its identity component has finite index. Simply connected-type semisimple G has connected G(R); reductive R-anisotropic groups have connected compact real points; conversely every compact linear Lie group is G′(R) for a reductive R-group G′.
Recorded result: real split and anisotropic tori
An R-split d-torus has real points (R^*)^d, while an R-anisotropic d-torus has compact connected real points; in dimension one it is R-isomorphic to SO(2), and topologically its real points form a circle.
Definition: nonarchimedean local field
The source uses “local field” for a completion with respect to a discrete nonarchimedean valuation, with valuation ring R, maximal ideal P=(π), and residue field R/P.
Theorem: affine Tits system for SL₂ over a local field
For SL₂(k), the explicitly defined Iwahori subgroup B and monomial normalizer N form a Tits system whose Weyl group is infinite dihedral on two simple reflections.
Recorded results: maximal compact subgroups and the rank-one building
For locally compact k, the two maximal proper parabolics of the affine BN-pair are the two conjugacy types of maximal compact subgroups of SL₂(k); the associated building is a tree, yielding the recorded freeness result for torsion-free discrete subgroups.
Definition: anisotropic kernel
For maximal k-split S with Z=C_G(S), the anisotropic kernel is the product of (Z,Z) with the maximal anisotropic k-torus in Z(Z); it is a maximal reductive k-anisotropic subgroup, determined up to k-isomorphism.
Definition: Tits index
The Tits index records the absolute Dynkin diagram, the subset of simple roots vanishing on S (the anisotropic kernel nodes), and the Galois action on the remaining diagram; roots in a common orbit have the same restriction to S.
Theorem/recorded result: Tits classification by anisotropic kernel and index
The source records Tits’s classification principle: a semisimple k-group is determined up to k-isomorphism by its anisotropic kernel (or derived part), its K-isomorphism class, and its Tits index, subject to the admissibility constraints on indexed diagrams.
Definition/result: reflection relative to a root vector
A reflection relative to nonzero α sends α to −α and fixes a hyperplane pointwise. If a finite spanning set Φ is reflection-stable, the reflection attached to α∈Φ is uniquely determined.
Definition: abstract root system
An abstract root system Φ⊂E is finite, spans E, omits 0, has only ±α on each root line, is stable under a reflection s_α for each root, and satisfies the integral Cartan condition s_α(β)−β∈Zα. Root-system isomorphisms preserve these integers; rank is dim E.
Definition/result: Weyl group and invariant inner product
W(Φ)=⟨s_α⟩ is finite and preserves an inner product for which s_α(β)=β−⟨β,α⟩α with Cartan integer ⟨β,α⟩=2(β,α)/(α,α).
Definition/results: bases, Weyl chambers and positive roots
A base Δ is a vector-space basis in which each root has integral coefficients all of one sign. Bases exist, W acts simply transitively on them, and they correspond to Weyl chambers; they define simple, positive and negative roots and the recorded two-root reduction properties.
Definition/result: length in a Weyl group
W is generated by simple reflections. The length of w is its minimal simple-reflection word length and equals the number of positive roots sent negative; a simple reflection permutes all positive roots except its own root.
Result: root subsystem generated by simple roots
For Δ′⊆Δ, the roots in span(Δ′) form a root system with base Δ′, whose Weyl group is the subgroup generated by the corresponding simple reflections.
Definition/result: irreducible root systems and their decomposition
Φ is irreducible when it cannot be split into mutually orthogonal proper root subsets. Every root system decomposes uniquely into irreducible components.
Classification: Dynkin classification and rank-two root systems
Irreducible reduced root systems are classified by Dynkin diagrams A,B,C,D,E,F,G; bonds encode Cartan integers and Coxeter orders. In rank two the possibilities are A₁×A₁, A₂, B₂ and G₂.
Theorem/result: automorphism group of a root system
Aut(Φ)=W⋊Γ, where Γ is the Dynkin-diagram automorphism group. For irreducible Φ, Γ is nontrivial only in types A_ℓ (ℓ≥2), D_ℓ and E_6.
Definition/result: abstract weight lattice and fundamental group table
Abstract weights λ satisfy integral Cartan pairings with all roots. They form a lattice Λ containing the root lattice Λ_r with finite index, with basis of fundamental dominant weights relative to Δ; the source records Λ/Λ_r for every irreducible type.
Definition/result: dominant weights and root order
A weight is dominant when its fundamental-weight coordinates are nonnegative. Each W-orbit contains a unique dominant weight; λ≥μ when λ−μ is a nonnegative integral sum of positive roots, and dominant λ satisfies λ≥wλ for all w∈W.
Definition/result: dual root system
The coroots α∨=2α/(α,α) form the dual root system Φ∨. Its Cartan matrix is transposed, its Weyl group is canonically W, and simple coroots are the dual linear functions characterized by value 2 on their root and 0 on the other fundamental weights.
Lemma: Tits root-chain lemma
If w(α)=β for simple roots α,β, then w factors into elements lying in rank-two parabolic Weyl subgroups along a chain of simple roots α=γ_0,…,γ_k=β, with each factor sending γ_i to γ_{i+1} and satisfying the stated rank-two support condition.