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Maximal tori of a smooth connected solvable group are conjugate

Statement

Assume the Axiom of Choice. Let $k$ be an algebraically closed field and let $G$ be a smooth connected solvable affine algebraic group over $k$ (Affine schemes and their coordinate rings, Smooth morphism of schemes, Group schemes of finite type over a field). Let $G_{u}$ be the largest smooth connected normal unipotent subgroup of $G$ and let $T \subseteq G$ be a maximal torus (Borel subgroups, maximal tori and Borel pairs, Groups of multiplicative type and tori). Then $G = G_{u} ⋊ T$ , every maximal torus of $G$ has dimension $\dim G - \dim G_{u}$ , and any two maximal tori of $G$ are conjugate by an element of $G (k)$ . Equivalently, every closed subgroup of multiplicative type of $G$ is conjugate into $T$ .

Facts & Assumptions

Given: The Axiom of Choice, an algebraically closed field $k$ , a smooth connected solvable affine $k$ -group $G$ , and a maximal torus $T \subseteq G$ .

[F1]

Assume AC. A smooth connected solvable affine group over an algebraically closed field is trigonalizable. The subgroup $G_{u}$ (the largest smooth connected normal unipotent subgroup) is the largest normal unipotent subgroup, $G / G_{u}$ is a smooth connected group of multiplicative type, hence a torus, and the extension $1 \to G_{u} \to G \to G / G_{u} \to 1$ splits: $G_{u}$ has a complement isomorphic to $G / G_{u}$ . (Lie-Kolchin: smooth connected solvable affine groups over algebraically closed fields are trigonalizable, Unipotent radicals of smooth connected trigonalizable groups over perfect fields have normal G_a series, Splitting trigonalizable extensions: algebraically closed fields and two perfect-field cases)

[F2]

Assume AC. For the trigonalizable group $G$ with $q : G \to D = G / G_{u}$ , the maximal diagonalizable subgroups are exactly the images $s (D)$ of the sections of $q$ , and any two are conjugate by an element of $G_{u} (k)$ ; under the present smooth connected hypotheses, $D$ is a torus and these are exactly the maximal tori. The supplier’s full diagonalizable-subgroup classification applies because $G_{u}$ is smooth connected, including in preimages with nonsmooth diagonalizable quotient. (Conjugacy of diagonalizable complements and maximal subgroups under smoothness hypotheses)

[F3]

A closed subgroup scheme that is both unipotent and diagonalizable is trivial; hence a closed subgroup of multiplicative type $S \subseteq G$ meets $G_{u}$ trivially and the exact kernel/image theorem identifies $S$ with its closed image $q (S)$ . (Group images are exact kernel quotients and preserve affine smooth connected properties) A closed subgroup scheme of a trigonalizable group is trigonalizable, and the preimage $q^{- 1} (H)$ of a closed subgroup $H \subseteq D$ has largest normal unipotent subgroup $G_{u}$ and quotient $H$ . (A subgroup that is both unipotent and diagonalizable is trivial, Unipotent groups are exactly the subgroups of some U_n, equivalently the groups with coconnected coordinate Hopf algebra, Trigonalizable algebraic groups)

[F4]

The split multiplication isomorphism identifies the underlying scheme with the product of the smooth affine groups $G_{u}$ and $D$ . Their associated reduced classical varieties are nonempty, so the product dimension theorem gives additivity: $\dim G = \dim G_{u} + \dim D$ , and the image of a section $s (D)$ is a closed subgroup isomorphic to the torus $D$ , hence a torus of dimension $\dim D$ . (Chain dimension and the empty-space convention, Dimensions add under products, Conjugacy of diagonalizable complements and maximal subgroups under smoothness hypotheses)

Proof

Given: The Axiom of Choice, an algebraically closed field $k$ , a smooth connected solvable affine $k$ -group $G$ , and a maximal torus $T \subseteq G$ .

1.1F1F2F4

By [F1] the group $G$ is trigonalizable, $D = G / G_{u}$ is a torus, and the extension $q : G \to D$ splits; by [F2] the images of the sections of $q$ are exactly the maximal tori of $G$ and any two of them are conjugate by an element of $G_{u} (k)$ . In particular the given maximal torus $T$ is the image $s_{T} (D)$ of a section and $\dim T = \dim D = \dim G - \dim G_{u}$ by [F4], and the multiplication morphism $G_{u} ⋊ D \to G$ , $(x, d) \mapsto x \cdot s_{T} (d)$ , is an isomorphism, so $G = G_{u} ⋊ T$ .

1.2F3

It remains to prove the equivalent statement for an arbitrary closed subgroup $S \subseteq G$ of multiplicative type. By [F3] $S \cap G_{u} = 1$ , so $q ∣_{S} : S \to q (S)$ is a closed immersion identifying $S$ with the closed subgroup $q (S) \subseteq D$ , and the preimage $G^{''} = q^{- 1} (q (S))$ is a closed subgroup scheme of $G$ , hence trigonalizable, with largest normal unipotent subgroup $G_{u}$ and quotient $q (S)$ .

2.1F2step 1.2

Both $(q ∣_{S})^{- 1} : q (S) \to S \subseteq G^{''}$ and $s^{''} = s_{T} ∣_{q (S)} : q (S) \to G^{''}$ are sections of the quotient $q^{''} = q ∣_{G^{''}} : G^{''} \to q (S)$ , The kernel $G_{u}$ remains smooth connected even if the subgroup $q (S)$ is nonsmooth, so the full classification/conjugacy domain of [F2] applies to this trigonalizable $G^{''}$ . Thus there is $u \in G_{u} (k)$ with $(q ∣_{S})^{- 1} = i n n (u) \circ s^{''}$ . Hence $S = i n n (u) (s_{T} (q (S))) \subseteq i n n (u) (T)$ , that is, $i n n (u)^{- 1} (S) \subseteq T$ : every closed subgroup of multiplicative type of $G$ is conjugate into the given maximal torus $T$ .

3.1step 1.1step 2.1∎

Collecting: the extension splits with complement the maximal torus $T$ , so $G = G_{u} ⋊ T$ ; all maximal tori have dimension $\dim G - \dim G_{u}$ and are pairwise conjugate by [step 1.1], and the equivalent conjugacy-into- $T$ statement for closed subgroups of multiplicative type is [step 2.1].

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Florian Herzig, Linear Algebraic Groups (University of Toronto lecture notes, 2013) (standard reference, not scraped)
J. S. Milne, Algebraic Groups (v2.00, 20 December 2015 author-hosted preliminary edition) (standard reference, not scraped)