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Highest Weights and Rational Representations of Split Reductive Groups

1 · Prerequisites

2 · Summary

This page develops the highest-weight theory of the rational representations of a split reductive group over an arbitrary field. It begins with the representation-theoretic foundations: the dictionary between rational representations and comodules, the contragredient representation Contragredient (dual) rational representation, tensor, Hom and exterior-power representations Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational, simple and semisimple representations Simple and semisimple rational representations, and the finiteness theorem Simple rational representations are finite-dimensional that every simple rational representation of an affine group scheme of finite type is finite-dimensional.

The linear-reductivity half of the page is characteristic-zero: the Lie algebra of a semisimple group is semisimple (The Lie algebra of a semisimple group in characteristic zero is semisimple), semisimple groups are perfect with no nontrivial characters (Semisimple groups are perfect and have no nontrivial characters), the Casimir operator of a rational representation is a module endomorphism (The Casimir element of a rational representation is an endomorphism of G-modules), and these combine into the linear reductivity theorem Semisimple groups in characteristic zero are linearly reductive and the complete-reducibility theorem Complete reducibility of rational modules in characteristic zero. Chevalley's line-stabilizer theorem Chevalley: every closed subgroup is a line stabilizer and the computation of Lie algebras of stabilizers Lie algebras of subspace stabilizers and Lie-stable subspaces supply the subgroup and ideal machinery, while The top exterior power detects stabilizers of a subspace reduces a subspace stabilizer to a line stabilizer.

The highest-weight half is characteristic-free. Weights, dominant weights and the dominance order are set up in Weights, dominant weights and the highest-weight order of a rational representation, primitive vectors in Primitive vectors for a Borel pair, and their root-group expansion and normalizer behaviour in Expansion of a root-group translate of a weight vector and The normalizer of the torus permutes weight spaces. Modules generated by a primitive vector have a one-dimensional top weight space and a simple quotient (Modules generated by a primitive vector); the induced coordinate module $E (λ)$ and its fixed line are The induced coordinate module E(lambda) and Primitive vectors of the induced coordinate module. Existence of a primitive vector of every dominant weight proceeds through the standard maximal parabolics Primitive vectors from standard maximal parabolics, fundamental-weight multiples Multiples of the fundamental weights are primitive weights in the semisimple case, tensor products Tensor products of primitive vectors, the semisimple case Every dominant weight of a split semisimple group is a primitive weight, the product with a torus Dominant characters of a torus times a split semisimple group are primitive weights and the descent along the central isogeny $Z (G)_{t} \times G_{d e r} \to G$ , where $Z (G)_{t}$ is the largest central torus, rather than the possibly nonreduced identity component of the centre, (Central characters and descent along a central isogeny, Every dominant character of a split reductive group is a highest weight). Simple representations of a split reductive group have a unique primitive line and highest weight (Simple rational representations have a unique highest weight) and are determined up to isomorphism by it (Simple modules with equal highest weight are isomorphic), which yields the classification Dominant weights classify the simple rational representations of a split reductive group. The remark The highest-weight classification does not imply semisimplicity in positive characteristic records that this classification says nothing about semisimplicity of extension modules, and the example companion highest-weights-and-rational-representations-of-split-reductive-groups-examples carries the ${S L}_{2}$ computations and the positive-characteristic counterexample.

The Axiom of Choice is carried only where the named suppliers use it, notably Cartier's smoothness theorem, the existence of the faithful finite-dimensional representation, the fppf quotient and the Noetherian finiteness inputs; the weight combinatorics and the root-group computations themselves are choice-free.

3 · Logical flowchart

4 · Definitions, theorems and proofs

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Power extension over a normal affine domain

Statement

Let $X$ be an integral affine scheme over a field $k$ whose coordinate ring $A = O (X)$ is a normal domain, i.e. integrally closed in its fraction field (Affine schemes and their coordinate rings, Global functions on Spec A recover A, Integral closure in an extension ring and integrally closed domains). Let $U \subseteq X$ be a dense open subscheme and let $f \in O (U)$ be such that $f^{d} \in O (X)$ for some integer $d \geq 1$ , where the rings are compared inside $O (X) \subseteq O (U) \subseteq Frac (A)$ . Then $f \in O (X)$ .

Facts & Assumptions

Given: An integral affine scheme $X$ with normal coordinate ring $A = O (X) = Γ (X, O_{X})$ , a dense open subscheme $U \subseteq X$ , an integer $d \geq 1$ and an element $f \in O (U)$ with $f^{d} \in O (X)$ , all inside $Frac (A)$ .

[F1]

Global sections of an affine scheme. The canonical map $A \to Γ (X, O_{X})$ is an isomorphism, so the global sections of $X$ are exactly $A = O (X)$ ; in particular $O (X)$ is a domain. (Global functions on Spec A recover A)

[F2]

Integrality criterion. Let $A \subseteq B$ be a ring extension and let $b \in B$ satisfy a monic polynomial with coefficients in $A$ ; then $b$ is integral over $A$ . (Integral closure in an extension ring and integrally closed domains)

[F3]

Integrally closed domain. Since $A$ is integrally closed in $Frac (A)$ , every element of $Frac (A)$ that is integral over $A$ already belongs to $A$ . (Integral closure in an extension ring and integrally closed domains)

Proof

technique · direct

1.1F1F2given

By hypothesis $f^{d} \in O (X) = A$ , so $f$ is a root of the monic polynomial $T^{d} - f^{d} \in A [T]$ ; hence $f$ is integral over $A$ .

2.1F1F3step 1.1∎

The element $f$ lies in $O (U) \subseteq Frac (A)$ , so step 1.1 and the integral closedness of $A$ give $f \in A = O (X)$ . This includes the cases $U = X$ and $d = 1$ trivially, and $f = 0$ is harmless because $Frac (A)$ is a field.

Remarks

The hypothesis $O (X) \subseteq O (U) \subseteq Frac (A)$ holds for every dense open subscheme of an integral affine scheme: restriction to the generic point injects $O (U)$ into $Frac (A)$ , and restriction from $X$ to $U$ identifies $A$ with a subring of $O (U)$ . This hypothesis is part of the statement because only the extension of rings, not the geometry of $U$ , is used.
The element $f$ need not be assumed nonzero: if $f^{d} = 0$ then $f = 0$ because $O (U) \subseteq Frac (A)$ is contained in a field, so the case $f = 0$ also satisfies the conclusion $f \in O (X)$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Trace forms of faithful representations of semisimple Lie algebras are nondegenerate

Statement

Let $h$ be a finite-dimensional semisimple Lie algebra over a field $k$ of characteristic $0$ and let $ρ : h \to g l (V)$ be a faithful finite-dimensional representation with trace form $B_{ρ} (x, y) = tr (ρ (x) ρ (y))$ (Trace form of a representation). Then $B_{ρ}$ is a nondegenerate symmetric invariant form on $h$ ; equivalently, its radical is zero (Trace forms are symmetric and invariant).

Facts & Assumptions

Given: A finite-dimensional Lie algebra $h$ over a characteristic-zero field $k$ with $rad (h) = 0$ , and a faithful finite-dimensional representation $ρ : h \to g l (V)$ . Write $B_{ρ} (x, y) = tr (ρ (x) ρ (y))$ for its trace form and $r = {x \in h : B_{ρ} (x, y) = 0 for every y \in h}$ for its radical.

[F1]

Symmetry and invariance. $B_{ρ}$ is bilinear and symmetric, and $B_{ρ} ([z, x], y) + B_{ρ} (x, [z, y]) = 0$ for all $x, y, z \in h$ (Trace forms are symmetric and invariant).

[F2]

Radicals of invariant forms are ideals. If $i$ is an ideal of a Lie algebra carrying a symmetric invariant bilinear form $B$ , then $i^{⊥}$ is an ideal; in particular $r = h^{⊥}$ is an ideal of $h$ (Orthogonal complements under invariant forms are ideals, Lie subalgebras, ideals, and center).

[F3]

Cartan's solvability criterion. Let $l \subseteq g l (V)$ be a finite-dimensional linear Lie algebra over a characteristic-zero field. If $tr (x y) = 0$ for all $x \in [l, l]$ and $y \in l$ , then $l$ is solvable (Cartan's solvability criterion).

[F4]

The solvable radical. The solvable radical $rad (h)$ is the largest solvable ideal of $h$ : it is solvable and contains every solvable ideal (Solvable radical), and semisimplicity means $rad (h) = 0$ (Simple, semisimple, and reductive Lie algebras).

[F5]

Faithfulness transfers solvability. If $ρ$ is injective and bracket preserving then $ρ (D^{j} r) = D^{j} ρ (r)$ for every $j$ , so $r$ is solvable as soon as the linear Lie algebra $ρ (r)$ is solvable; ideals and quotients of a semisimple algebra are again semisimple (Ideals and quotients of semisimple Lie algebras).

Proof

technique · direct

1.1F1given

The trace form $B_{ρ}$ is symmetric and invariant, and its radical $r$ is a linear subspace of $h$ .

2.1F2step 1.1

The radical $r = h^{⊥}$ is an ideal of $h$ : it is the orthogonal complement of the ideal $h$ , and orthogonal complements of ideals under symmetric invariant forms are ideals.

3.1givenstep 2.1algebra

Since $r$ is an ideal by step 2.1, $[r, r] \subseteq r$ . Thus $B_{ρ} (x, y) = 0$ for every $x \in [r, r]$ and every $y \in h$ , by the definition of the radical $r$ .

4.1F3step 3.1

The linear Lie algebra $ρ (r) \subseteq g l (V)$ is solvable. Its derived algebra is $[ρ (r), ρ (r)] = ρ ([r, r])$ , and for $x \in [r, r]$ and $y \in r$ step 3.1 gives $tr (ρ (x) ρ (y)) = B_{ρ} (x, y) = 0$ ; Cartan's criterion therefore makes $ρ (r)$ solvable.

5.1step 4.1

The algebra $r$ is solvable. The representation $ρ$ is injective and bracket preserving, so $ρ (D^{j} r) = D^{j} ρ (r)$ for every $j$ ; since $ρ (r)$ is solvable by step 4.1, [F5] gives that $r$ is solvable: some $D^{j} ρ (r)$ is zero, hence $D^{j} r = 0$ .

6.1F4step 2.1step 5.1∎

By step 5.1 the radical $r$ is a solvable ideal of $h$ , so $r \subseteq rad (h) = 0$ because the solvable radical is the largest solvable ideal and $h$ is semisimple. Hence $r = 0$ , that is, $B_{ρ}$ is nondegenerate; together with step 1.1 this proves the lemma.

Remarks

Semisimplicity of $h$ is used exactly once, at step 6.1, through $rad (h) = 0$ : any solvable ideal lies in the radical.
The faithfulness of $ρ$ is used exactly at step 5.1 to transfer solvability from the linear algebra $ρ (r)$ back to $r$ ; a nonfaithful representation can have degenerate trace form.
Characteristic zero enters only through Cartan's solvability criterion at step 4.1.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Contragredient (dual) rational representation

Definition

Let $(V, r)$ be a finite-dimensional rational representation of an affine group scheme $G$ over $k$ (Rational representations and comodules of an affine group scheme). The contragredient representation $r^{\lor}$ is the representation on the algebraic dual $V^{*} = {Hom}_{k} (V, k)$ (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ) defined by $(r^{\lor} (g) f) (v) = f (r (g)^{- 1} v)$ for $g \in G (R)$ , $f \in V_{R}^{*}$ and $v \in V_{R}$ . When $V$ is finite-dimensional, $r^{\lor}$ is a rational representation and its comodule is the dual comodule of $(V, r)$ ; moreover the weights of $(V^{*}, r^{\lor})$ are the negatives of the weights of $(V, r)$ .

Remarks

Left action. For $R$ -points $g, h$ and $f \in V_{R}^{*}$ one has $(r^{\lor} (g) (r^{\lor} (h) f)) (v) = (r^{\lor} (h) f) (r (g)^{- 1} v) = f (r (h)^{- 1} r (g)^{- 1} v) = f (r (g h)^{- 1} v) = (r^{\lor} (g h) f) (v)$ for all $v \in V_{R}$ , since $r$ is a representation and $(g h)^{- 1} = h^{- 1} g^{- 1}$ ; hence $r^{\lor}$ is a left action by $R$ -linear automorphisms of $V_{R}^{*}$ . The inverse in the formula is what makes the action a left action, and it is also the reason that the contragredient of a contragredient recovers the original representation.
Rationality in the finite-dimensional case. Choose a basis $e_{1}, \dots, e_{n}$ of $V$ , its dual basis $e_{1}^{*}, \dots, e_{n}^{*}$ , and write $ρ (e_{j}) = \sum_{i} e_{i} \otimes a_{i j}$ . The dual coaction is $ρ^{\lor} (e_{i}^{*}) = \sum_{j} e_{j}^{*} \otimes S (a_{i j}),$ where $S$ is the antipode of $O (G)$ . Evaluating at $g \in G (R)$ gives the transpose of $r_{R} (g)^{- 1}$ , so its action is the displayed formula. The inverse and transpose matrix identities give the group law naturally in $R$ , hence the comodule identities by the representation/comodule dictionary. All matrix entries are regular functions on $G$ .
Weights. If $V$ is finite-dimensional and $T$ is a diagonalizable group acting on $V$ with weight spaces $V_{χ}$ , then the dual basis of a basis of $V_{χ}$ spans the weight space $(V^{*})^{- χ}$ : for $f \in (V^{*})^{- χ}$ and $v \in V_{χ}$ the pairing is compatible with the dual action, so the weights of $V^{*}$ are exactly the $- χ$ with $V_{χ} \neq 0$ . This is the fact used for the contragredient of a simple module.
Infinite-dimensional case. For arbitrary $V$ , the formula $f \mapsto f \circ r_{k} (g)^{- 1}$ defines an action of the abstract group $G (k)$ on the full algebraic dual. It need not be rational and need not extend to the module $V^{*} \otimes_{k} R$ for every $k$ -algebra $R$ . For example, let $G = G_{m}$ , $V = ⨁_{n \geq 0} k e_{n}$ with $e_{n}$ of weight $n$ , and $f (e_{n}) = 1$ . Over $R = k [t, t^{- 1}]$ , precomposition by the universal point gives values $t^{- n}$ , which cannot lie in $V^{*} \otimes_{k} R$ : values of any element of that tensor product span a finite-dimensional $k$ -subspace of $R$ . Thus the rational contragredient above is stated for finite-dimensional representations, exactly the range used by its consumers.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Simple and semisimple rational representations

Definition

Let $k$ be a field, let $G$ be an affine group scheme of finite type over $k$ , and let $(V, r)$ be a rational representation of $G$ , with subrepresentations the subcomodules of $V$ (Rational representations and comodules of an affine group scheme). The representation $(V, r)$ is simple (or irreducible) if $V \neq 0$ and the only subrepresentations of $V$ are $0$ and $V$ . It is semisimple (or completely reducible) if $V$ is an internal direct sum of simple subrepresentations, $V = ⨁_{i \in I} S_{i}, S_{i} \subseteq V simple;$ the zero representation is semisimple, being the empty direct sum.

Remarks

Terminology. Simple and semisimple representations are traditionally called irreducible and completely reducible when regarded as representations; the two pairs of words are synonyms here, as in the source.
Finite dimensionality. Every simple rational representation of an affine finite-type group scheme is finite-dimensional; this is proved on the same page and is not part of the definition.
Subrepresentations. Under the correspondence of the cited definition, subrepresentations are exactly the subcomodules, so simplicity and semisimplicity can be checked on comodules; no smoothness of $G$ is required, and no choice principle is used in the definition.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Lie algebras of subspace stabilizers and Lie-stable subspaces

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be an affine group scheme of finite type over a field $k$ with Lie algebra $g = Lie (G)$ (The Lie algebra of a group scheme), let $(V, r)$ be a rational representation (Rational representations and comodules of an affine group scheme), and let $W \subseteq V$ be a subspace with scheme-theoretic stabilizer ${Stab}_{G} (W)$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers). Then: (a) $Lie ({Stab}_{G} (W)) = {x \in g : x W \subseteq W}$ (Milne 10.31); (b) if moreover $k$ has characteristic $0$ , $G$ is connected and smooth, and $g W \subseteq W$ , then $W$ is $G$ -stable. In particular a subspace of a finite-dimensional representation of a connected semisimple group in characteristic zero is $G$ -stable if and only if it is stable under $g$ .

Facts & Assumptions

Given: An affine group scheme $G$ of finite type over $k$ with Lie algebra $g$ (The Lie algebra of a group scheme), a rational representation $(V, r)$ with differential $d r : g \to {g l}_{V}$ , a subspace $W \subseteq V$ , and the scheme-theoretic stabilizer ${Stab}_{G} (W)$ with $R$ -points ${g \in G (R) : r (g) W_{R} = W_{R}}$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers).

[F1]

The universal stabilizer equations. Put $A = O (G)$ and $Q = V / W$ . The representation coaction and its inverse coaction are $ρ : V \to V \otimes A$ and $(id \otimes S) ρ$ . Under AC, $Q$ has a basis by Every vector space has a basis, so its coordinate functionals detect zero in $Q \otimes_{k} R$ for every $k$ -algebra $R$ . Applying these functionals to the images of $w \in W$ under both coactions gives coefficient equations in $A$ for $r_{R} (g) W_{R} \subseteq W_{R}$ and $r_{R} (g)^{- 1} W_{R} \subseteq W_{R}$ . Together they express equality and cut out a closed subgroup scheme of $G$ . Its coordinate algebra is a quotient of $A$ , hence finitely generated over $k$ by An affine scheme of finite type over a field has a finitely generated coordinate ring; thus the stabilizer is of finite type, even when $V$ is infinite-dimensional. (Rational representations and comodules of an affine group scheme, Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers, Fibre product of schemes)

[F2]

The differential action. A dual-number point $e^{ε X}$ acts on $V [ε]$ by $1 + ε d r (X)$ , with inverse $1 - ε d r (X)$ . This follows directly by evaluating the representation coaction at a point reducing to the identity. For finite-dimensional $V$ , it is the matrix calculation $Lie ({GL}_{n}) = M_{n} (k)$ of The Lie algebra of the general linear group; the same coaction calculation works for arbitrary $V$ without treating its automorphism functor as a finite-type scheme. (The Lie algebra of a group scheme, The tangent space at the identity is a vector space, and Lie is a functor)

[F3]

Left exactness of $Lie$ . For algebraic subgroups $H_{1}, H_{2} \subseteq G$ with fibre product over a morphism, $Lie$ commutes with the fibre product: in particular the Lie algebra of the pullback of a closed subgroup under a morphism is the fibre product of the Lie algebras, and $Lie (H_{1} \cap H_{2}) = Lie (H_{1}) \cap Lie (H_{2})$ (see (a)); moreover if $Lie (H) = Lie (G)$ , $H$ is smooth and $G$ is connected, then $H = G$ (see (b)) (The Lie functor: exactness, fixed points and generation, The tangent space at the identity is a vector space, and Lie is a functor).

[F4]

Cartier's theorem. In characteristic $0$ every affine group scheme of finite type over $k$ is smooth (Cartier's theorem: affine group schemes in characteristic zero are smooth; AC is used here and is inherited by this item).

Proof

technique · direct

1.1F1F2given

The subgroup $H = {Stab}_{G} (W)$ is represented by the closed coefficient equations of [F1]. A point $e^{ε X} \in G (k [ε])$ acts by $1 + ε d r (X)$ by [F2], so it carries $w_{0} + ε w_{1}$ to $w_{0} + ε (w_{1} + d r (X) w_{0})$ . This belongs to $W [ε]$ for every $w_{0}, w_{1} \in W$ exactly when $d r (X) W \subseteq W$ . In that case the inverse $1 - ε d r (X)$ also preserves $W [ε]$ , so the condition is equality of submodules, as required by the stabilizer functor. This proves the criterion without any dimensional restriction on $V$ .

2.1F2F3step 1.1

Part (a). The Lie algebra of the closed subgroup $H$ consists of its dual-number points reducing to the identity. The inclusion $H ↪ G$ injects those points into $g$ by [F3]; step 1.1 identifies its image with ${X \in g : d r (X) W \subseteq W}$ . Hence $Lie (H) = {X \in g : X W \subseteq W}$ , with the differential action understood as in the Statement.

3.1F3F4step 2.1

Part (b). Assume $char k = 0$ , $G$ connected and smooth, and $g W \subseteq W$ . By step 2.1, $Lie ({Stab}_{G} (W)) = g$ ; the stabilizer is an affine group scheme of finite type, so it is smooth by Cartier's theorem; and the Lie-exactness criterion for connected groups now gives ${Stab}_{G} (W) = G$ , that is, $W$ is $G$ -stable.

4.1F4step 2.1step 3.1∎

The particular case. If $G$ is connected semisimple in characteristic $0$ , then $G$ is smooth by Cartier's theorem, and step 3.1 applies: $g W \subseteq W$ implies that $W$ is $G$ -stable. Conversely, if $W$ is $G$ -stable then ${Stab}_{G} (W) = G$ and step 2.1 gives $g W = Lie ({Stab}_{G} (W)) W \subseteq W$ . Hence $W$ is $G$ -stable if and only if $g W \subseteq W$ .

Remarks

In positive characteristic the implication (b) can fail: $G$ need not be smooth, and $Lie ({Stab}_{G} (W)) = g$ does not force ${Stab}_{G} (W) = G$ ; this is why both characteristic $0$ and Cartier's theorem appear in the statement.
The equality of part (a) is Milne 10.31; the extra hypothesis $g W \subseteq W$ in part (b) is exactly $Lie ({Stab}_{G} (W)) = Lie (G)$ by part (a).

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Simple rational representations are finite-dimensional

Statement

Let $G$ be an affine group scheme of finite type over a field $k$ . Then every simple rational representation $(V, r)$ of $G$ is finite-dimensional (Rational representations and comodules of an affine group scheme, Simple and semisimple rational representations).

Facts & Assumptions

Given: An affine group scheme $G$ of finite type over $k$ with coordinate Hopf algebra $A = O (G)$ , and a simple rational representation $(V, r)$ , so $V \neq 0$ and the only subrepresentations of $V$ are $0$ and $V$ (Simple and semisimple rational representations).

[F1]

Finite-dimensional subcomodules. For every finite subset $S$ of an $A$ -comodule $M$ there is a finite-dimensional subcomodule $N \subseteq M$ containing $S$ ; consequently $M$ is the directed union of its finite-dimensional subcomodules (Every element of a comodule lies in a finite-dimensional subcomodule).

[F2]

Subrepresentations are subcomodules. Under the comodule dictionary, subrepresentations of $V$ correspond to subcomodules, and a nonzero subcomodule is a nonzero subrepresentation (Rational representations and comodules of an affine group scheme).

Proof

technique · direct

1.1given

Since $V \neq 0$ , choose a nonzero vector $v \in V$ .

2.1F1step 1.1

By [F1] there is a finite-dimensional subcomodule $W \subseteq V$ containing $v$ .

3.1F2givenstep 2.1∎

By [F2] the subspace $W$ is a subrepresentation of $V$ ; it is nonzero because $v \in W$ . Since $V$ is simple, its only subrepresentations are $0$ and $V$ , so $W = V$ . Hence $V$ is finite-dimensional, as claimed.

Remarks

The only input is Milne 4.8: the comodule structure makes every element lie in a finite-dimensional subcomodule, and a simple module cannot have a nonzero proper submodule.
No hypothesis on $k$ beyond being a field is used, and no choice principle: the finite-dimensional subcomodule is produced from finitely many coefficients of $ρ (v)$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational

Statement

Let $G$ be an affine group scheme of finite type over a field $k$ with coordinate Hopf algebra $A = O (G)$ , and let $(V, r)$ and $(W, s)$ be finite-dimensional rational representations with comodule maps $ρ_{V} : V \to V \otimes_{k} A$ , $ρ_{W} : W \to W \otimes_{k} A$ (Rational representations and comodules of an affine group scheme). (a) The formula $c (v \otimes w) = \sum_{i, j} v_{i} \otimes w_{j} \otimes a_{i} b_{j}$ for $ρ_{V} (v) = \sum_{i} v_{i} \otimes a_{i}$ and $ρ_{W} (w) = \sum_{j} w_{j} \otimes b_{j}$ defines the unique comodule structure on $V \otimes_{k} W$ whose associated rational representation is $g \cdot (v \otimes w) = g \cdot v \otimes g \cdot w$ . (b) The space ${Hom}_{k} (V, W)$ carries a rational representation with $(g \cdot f) (v) = g \cdot f (g^{- 1} \cdot v)$ , and the canonical $k$ -linear map $V^{*} \otimes_{k} W \to {Hom}_{k} (V, W)$ , $ξ \otimes w \mapsto (v \mapsto ξ (v) w)$ , where $V^{*}$ is the contragredient (Contragredient (dual) rational representation, Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ), is an isomorphism of rational representations. (c) For every $d \geq 0$ the exterior power $Λ^{d} V$ carries a rational representation with $g \cdot (v_{1} \land \dots \land v_{d}) = g \cdot v_{1} \land \dots \land g \cdot v_{d}$ , and for every $k$ -algebra $R$ the induced map on $Λ_{R}^{d} (V_{R})$ is $Λ_{R}^{d} (r_{R} (g))$ under the identification of the two $R$ -modules by the common wedge basis (The $k$ th exterior power as the tensor-power quotient by repeated-vector relations, Increasing-index wedges of a basis form a basis of $Λ^{k} V$ ).

Facts & Assumptions

Given: An affine group scheme $G$ of finite type over $k$ with coordinate Hopf algebra $(A, Δ, ε, S)$ , finite-dimensional rational representations $(V, r)$ , $(W, s)$ with comodule maps $ρ_{V}$ , $ρ_{W}$ as above, and the contragredient $V^{*}$ of $(V, r)$ .

[F1]

Comodule dictionary. $ρ \mapsto r$ with $r_{R} (g) (v \otimes 1) = ({id}_{V} \otimes g) ρ (v)$ (extended $R$ -linearly) is a bijection from comodule structures on $V$ to rational representations on $V$ , natural in $V$ , and it maps subcomodules to subrepresentations (Rational representations and comodules of an affine group scheme, Rational representations of an affine group scheme are comodules of its coordinate Hopf algebra).

[F2]

Comodule and Hopf axioms. $(ρ \otimes id) ρ = (id \otimes Δ) ρ$ and $(id \otimes ε) ρ = id$ , $Δ$ and $ε$ are $k$ -algebra homomorphisms, and $(Δ \otimes id) Δ = (id \otimes Δ) Δ$ (Commutative Hopf algebras over a field, Rational representations and comodules of an affine group scheme).

[F3]

Tensor products. The decomposable tensors $v \otimes w$ span $V \otimes_{k} W$ as an abelian group, and every $k$ -bilinear map from $V \times W$ to an abelian group induces a unique group homomorphism from $V \otimes_{k} W$ . For vector spaces over the commutative field $k$ , the quotient presentation also gives the scalar action $λ (v \otimes w) = (λ v) \otimes w$ ; its well-definedness follows because scaling the first variable carries each additivity and balancing relation to another defining relation. Iterated tensor products inherit this action, with scalars movable between factors by the balancing relation (The tensor product $M \otimes_{R} N$ from the additive group underlying the free $Z$ -module on $M \times N$ , elementary tensors, and finite tensor sums, Universal property of the tensor product for balanced maps into abelian groups).

[F4]

Contragredient. For finite-dimensional $V$ the dual $V^{*}$ is a rational representation with $(r^{\lor} (g) f) (v) = f (r (g)^{- 1} v)$ for $R$ -points (Contragredient (dual) rational representation).

[F5]

Linear algebra of $Hom$ . For finite-dimensional $V$ , the canonical map $V^{*} \otimes_{k} W \to {Hom}_{k} (V, W)$ , $ξ \otimes w \mapsto (v \mapsto ξ (v) w)$ , is a $k$ -linear isomorphism (For finite-dimensional $V$ , the canonical map $V^{*} \otimes_{F} W \to {Hom}_{F} (V, W)$ is an isomorphism, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ).

[F6]

Scalar extension of a finite basis. If $e_{1}, \dots, e_{n}$ is a $k$ -basis of $V$ with coordinate functionals $e_{i}^{*}$ , then $V_{R} = V \otimes_{k} R$ is free over $R$ with basis $e_{i} \otimes 1$ . Indeed, the maps $(r_{i}) \mapsto \sum_{i} e_{i} \otimes r_{i}$ and $v \otimes r \mapsto (e_{i}^{*} (v) r)_{i}$ are inverse; the second is induced by the $k$ -bilinear tensor map of [F3] and is $R$ -linear for the action on the second factor (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ).

[F7]

Exterior algebra bases over a ring. If $F$ is a finite free module over a commutative ring $R$ with ordered basis $e_{1}, \dots, e_{n}$ , its degree- $d$ exterior power $Λ_{R}^{d} (F)$ has $R$ -basis $e_{i_{1}} \land \dots \land e_{i_{d}}$ for $i_{1} < \dots < i_{d}$ ; the basis is empty and the module is zero when $d > n$ (Exterior Algebra Of A Finite Free Module, Exterior Algebra Basis Monomials).

[F8]

Exterior-power basis over the field. If $e_{1}, \dots, e_{n}$ is an ordered basis of $V$ , then the increasing wedges $e_{I}$ form a $k$ -basis of $Λ_{k}^{d} V$ for $1 \leq d \leq n$ (Increasing-index wedges of a basis form a basis of $Λ^{k} V$ ).

[F9]

Exterior-power universal property. Every alternating $k$ -multilinear map out of $V^{d}$ factors uniquely through $Λ_{k}^{d} V$ (Exterior powers represent alternating multilinear maps and are unique up to unique isomorphism).

Proof

technique · direct

1.1F3given

The right-hand side of (a) is $k$ -bilinear in $(v, w)$ , since both comodule maps are $k$ -linear and scalars move between tensor factors over $k$ ; [F3] therefore gives a unique additive group homomorphism $c : V \otimes_{k} W \to V \otimes_{k} W \otimes_{k} A$ . This homomorphism is $k$ -linear: on every decomposable tensor, $c (λ (v \otimes w)) = c ((λ v) \otimes w) = λ c (v \otimes w)$ by the formula, and decomposable tensors generate the source additively.

2.1F2step 1.1

Counit axiom. Applying $id \otimes ε$ to $c (v \otimes w) = \sum v_{i} \otimes w_{j} \otimes a_{i} b_{j}$ and using that $ε$ is multiplicative and $(id \otimes ε) ρ_{V} = {id}_{V}$ , $(id \otimes ε) ρ_{W} = {id}_{W}$ gives $\sum v_{i} \otimes w_{j} ε (a_{i}) ε (b_{j}) = v \otimes w$ , so $(id \otimes ε) c = {id}_{V \otimes W}$ .

2.2F2step 1.1

Coassociativity. Write the comultiplications in Sweedler notation, $ρ_{V} (v) = \sum v_{(1)} \otimes v_{(2)}$ , $ρ_{W} (w) = \sum w_{(1)} \otimes w_{(2)}$ . Then $(c \otimes id) c (v \otimes w) = \sum v_{(1) (1)} \otimes w_{(1) (1)} \otimes v_{(1) (2)} w_{(1) (2)} \otimes v_{(2)} w_{(2)}$ , while $(id \otimes Δ) c (v \otimes w) = \sum v_{(1)} \otimes w_{(1)} \otimes v_{(2) (1)} w_{(2) (1)} \otimes v_{(2) (2)} w_{(2) (2)}$ ; the two sums agree after rewriting the first three tensor factors with the coassociativity identities for $ρ_{V}$ and $ρ_{W}$ and using multiplicativity of $Δ$ and commutativity of $A$ in the last two factors. Hence $(c \otimes id) c = (id \otimes Δ) c$ .

2.3F1step 1.1

The associated representation. By [F1] the representation associated with $c$ acts on $R$ -points by sending $v \otimes w$ to $(id \otimes {ev}_{g}) c (v \otimes w) = \sum v_{i} \otimes w_{j} a_{i} (g) b_{j} (g)$ , and this equals $(\sum_{i} v_{i} a_{i} (g)) \otimes (\sum_{j} w_{j} b_{j} (g)) = (g \cdot v) \otimes (g \cdot w)$ because the two $R$ -valued sums are exactly the actions of $g$ on $v$ and on $w$ under [F1]. This proves (a).

3.1F4F5step 2.3

Hom is rational. For a $k$ -algebra $R$ and $g \in G (R)$ , the tensor product of the rational representations $V^{*}$ and $W$ acts on $V^{*} \otimes_{k} W$ by $g \cdot (ξ \otimes w) = (g \cdot ξ) \otimes (g \cdot w)$ by step 2.3 applied to the pair $(V^{*}, W)$ , and $(g \cdot ξ) (v) = ξ (g^{- 1} v)$ by [F4]. Under the isomorphism $V^{*} \otimes_{k} W \to {Hom}_{k} (V, W)$ of [F5], the element $ξ \otimes w$ corresponds to the map $f (v) = ξ (v) w$ , and $g \cdot (ξ \otimes w)$ corresponds to $v \mapsto ξ (g^{- 1} v) g \cdot w = g \cdot (ξ (g^{- 1} v) w) = g \cdot f (g^{- 1} v)$ ; the transport is therefore the action $(g \cdot f) (v) = g \cdot f (g^{- 1} v)$ on ${Hom}_{k} (V, W)$ , which is thus a rational representation isomorphic to $V^{*} \otimes_{k} W$ . This proves (b).

3.2F1F2F3F6F7F8F9step 2.2

Exterior powers are rational and commute with scalar extension. For $d = 0$ , the exterior power is $k$ with the trivial action, and its base change is $R$ with the identity map. For $d \geq 1$ , write $ρ_{V} (v) = \sum_{i} v_{i} \otimes a_{i}$ and define $c_{d} : Λ_{k}^{d} V \to Λ_{k}^{d} V \otimes_{k} A$ by $c_{d} (v_{1} \land \dots \land v_{d}) = \sum_{i_{1}, \dots, i_{d}} v_{1, i_{1}} \land \dots \land v_{d, i_{d}} \otimes a_{1, i_{1}} \dots a_{d, i_{d}}$ , where $ρ_{V} (v_{j}) = \sum_{i} v_{j, i} \otimes a_{j, i}$ . This formula is alternating in $v_{1}, \dots, v_{d}$ : if two inputs coincide, terms with distinct corresponding indices cancel in pairs by $x \land y = - y \land x$ and commutativity of $A$ , and equal-index terms vanish; hence [F9] makes it well-defined. The counit and coassociativity axioms follow from multiplicativity of $ε$ , coassociativity of $ρ_{V}$ , and multiplicativity of $Δ$ , as in steps 2.1--2.2. Thus $Λ_{k}^{d} V$ is a comodule, and its associated action sends each decomposable wedge to the wedge of the actions by [F1] and the computation of step 2.3 with $d$ factors. Now choose an ordered basis $e_{1}, \dots, e_{n}$ of $V$ and write $e_{I}$ for its increasing-index wedges. For $1 \leq d \leq n$ , [F8] gives that the $e_{I}$ form a $k$ -basis of $Λ_{k}^{d} V$ ; if $d > n$ , expanding decomposable wedges in the $e_{i}$ gives only repeated-index wedges, which vanish in the quotient defining $Λ_{k}^{d} V$ , so that space is zero. By [F6], the $e_{i} \otimes 1$ form an $R$ -basis of $V_{R}$ , and [F7] gives the matching $R$ -basis $e_{I}^{R}$ of $Λ_{R}^{d} (V_{R})$ (or zero for $d > n$ ). The alternating $k$ -multilinear map $(v_{1}, \dots, v_{d}) \mapsto (v_{1} \otimes 1) \land \dots \land (v_{d} \otimes 1)$ induces a map $Λ_{k}^{d} V \to Λ_{R}^{d} (V_{R})$ by [F9]; multiplying its values by $r \in R$ gives a $k$ -balanced map $Λ_{k}^{d} V \times R \to Λ_{R}^{d} (V_{R})$ , so [F3] induces a group homomorphism $β_{d} : (Λ_{k}^{d} V) \otimes_{k} R \to Λ_{R}^{d} (V_{R})$ . It is $R$ -linear because $β_{d} (x \otimes a r) = a β_{d} (x \otimes r)$ on elementary tensors, which generate additively. It sends $e_{I} \otimes 1$ to $e_{I}^{R}$ , hence is an isomorphism by the two basis descriptions, with both sides zero when $d > n$ . For every $g \in G (R)$ , the tensor-power map of $r_{R} (g)$ preserves the ideal generated by the squares $u \otimes u$ in the exterior algebra of [F7], so descends to $Λ_{R}^{d} (r_{R} (g))$ ; both it and the base-changed action send $e_{I} \otimes 1$ to $r_{R} (g) (e_{i_{1}} \otimes 1) \land \dots \land r_{R} (g) (e_{i_{d}} \otimes 1)$ . They therefore agree on the basis and under $β_{d}$ . This proves (c).

4.1step 3.2∎

The degree-zero and positive-degree cases above establish the stated exterior-power action and its base change for every $d \geq 0$ .

Remarks

The lemma isolates the two structural facts that Milne's proof of 22.40 uses when it applies the codimension-one splitting hypothesis to the subspace $V_{1} = {f : f ∣_{W} = a {id}_{W}}$ of ${Hom}_{k} (V, W)$ : that tensor products of finite-dimensional rational representations are rational, and that ${Hom}_{k} (V, W)$ with $(g \cdot f) (v) = g \cdot f (g^{- 1} v)$ is rational (isomorphic to $V^{*} \otimes W$ ).
Part (c) is the input for applying the exterior-power stabilizer lemma to a rational representation: it makes the action on $Λ^{d} V$ a rational representation, so that its scheme-theoretic stabilizers are defined.
For infinite-dimensional $V$ the map $V^{*} \otimes_{k} W \to {Hom}_{k} (V, W)$ is injective but not surjective in general, which is why the finite-dimensionality hypothesis is part of the statement.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Complete reducibility reduces to splitting codimension-one simple submodules

Statement

Let $G$ be an algebraic group over a field $k$ with $X (G) = {Hom}_{k} (G, G_{m}) = 0$ , so that every one-dimensional rational representation of $G$ is trivial (Rational representations and comodules of an affine group scheme, Simple and semisimple rational representations). For a possibly nonaffine $G$ , a finite-dimensional rational representation here means a morphism $G \to {GL}_{V}$ , with subrepresentations the invariant subspaces; this agrees with the cited comodule definition when $G$ is affine. Then the following conditions are equivalent: (a) every finite-dimensional rational representation of $G$ is semisimple; (b) every subrepresentation $W$ of codimension one in a finite-dimensional representation $V$ is a direct summand; (c) every simple subrepresentation $W$ of codimension one in a finite-dimensional representation $V$ is a direct summand.

Facts & Assumptions

Given: A field $k$ , an algebraic group $G$ over $k$ with $X (G) = 0$ , and the notions of simple and semisimple rational representations and of subrepresentations (Simple and semisimple rational representations).

[F1]

Semisimplicity. A rational representation $V$ is semisimple when $V = ⨁_{i \in I} S_{i}$ is an internal direct sum of simple subrepresentations; subrepresentations are the invariant subspaces (equivalently subcomodules when $G$ is affine), and the image of a subrepresentation under an equivariant map is a subrepresentation (Simple and semisimple rational representations, Rational representations and comodules of an affine group scheme).

[F2]

Hom representations. For finite-dimensional rational representations $V, W$ , the space ${Hom}_{k} (V, W)$ with $(g \cdot f) (v) = g \cdot f (g^{- 1} v)$ is a finite-dimensional rational representation, isomorphic to $V^{*} \otimes_{k} W$ (Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational). For a nonaffine $G$ , the same formula is rational directly: in finite bases its matrix entries are products of the regular entries of the two representation matrices and of the inverse representation matrix, and the group law follows by substitution. Thus it defines a morphism $G \to GL ({Hom}_{k} (V, W))$ without an affineness hypothesis.

[F3]

Characters of one-dimensional representations. A rational representation of $G$ on a one-dimensional $k$ -space is given by a morphism $G \to G_{m}$ , that is, by an element of $X (G)$ ; since $X (G) = 0$ , such a representation is trivial, and a nonzero vector of it is fixed by every $R$ -point of $G$ (Rational representations and comodules of an affine group scheme, given).

Proof

technique · direct

1.1given

(b) $\Rightarrow$ (c) is immediate: a simple subrepresentation of codimension one is in particular a subrepresentation of codimension one.

1.2F1given

(a) $\Rightarrow$ (b). Since $V$ is finite-dimensional, its direct-sum decomposition has finitely many simple summands. Induct on their number. The zero-summand case is immediate. Write $V = S \oplus V^{'}$ with $S$ simple and $V^{'}$ a sum of fewer simple subrepresentations, and let $q : V \to V^{'}$ be the projection. For a subrepresentation $W \subseteq V$ , simplicity gives either $W \cap S = S$ or $W \cap S = 0$ . In the first case, $W = S \oplus (W \cap V^{'})$ ; induction splits $W \cap V^{'}$ in $V^{'}$ , hence splits $W$ in $V$ . In the second case, $q ∣_{W}$ is injective and $q (W)$ is a subrepresentation of $V^{'}$ ; induction gives $V^{'} = q (W) \oplus C^{'}$ for some subrepresentation $C^{'}$ . Set $C = S \oplus C^{'}$ . Every $v \in V$ differs from some $w \in W$ by an element of $C$ , because the $q (W)$ component of $q (v)$ has a unique lift through the injective map $q ∣_{W}$ . If $w \in W \cap C$ , then $q (w) \in q (W) \cap C^{'} = 0$ , so $w \in W \cap S = 0$ . Thus $V = W \oplus C$ in this case as well.

1.3given

(c) $\Rightarrow$ (b), induction on $n = \dim V$ . Assume (c) and let $W \subseteq V$ be a subrepresentation of codimension one in an $n$ -dimensional $V$ . If $W = 0$ then $V = 0 \oplus V$ ; if $W$ is simple then (c) applies; so suppose $W \neq 0$ is not simple. Then $W$ has a nonzero proper subrepresentation, and a maximal proper subrepresentation $W^{'}$ of $W$ exists and has $W / W^{'}$ simple, since any strictly increasing chain of proper subrepresentations of the finite-dimensional $W$ has length at most $\dim W$ . The quotient $V / W^{'}$ has dimension $n - \dim W^{'} < n$ when $W^{'} \neq 0$ , and $W / W^{'}$ is a simple subrepresentation of codimension one in $V / W^{'}$ , so (c) gives $V / W^{'} = W / W^{'} \oplus V^{'} / W^{'}$ for a subrepresentation $V^{'} \subseteq V$ containing $W^{'}$ ; then $\dim V^{'} / W^{'} = 1$ , so $\dim V^{'} = \dim W^{'} + 1$ . If $W^{'}$ is simple, (c) splits the pair $W^{'} \subseteq V^{'}$ ; if $W^{'}$ is not simple, then $\dim V^{'} < n$ and the induction hypothesis (b) applied to $V^{'}$ splits the pair $W^{'} \subseteq V^{'}$ . In both cases $V^{'} = W^{'} \oplus L$ for a one-dimensional subrepresentation $L$ . Now $W \cap V^{'} = W^{'}$ because $W / W^{'} \cap V^{'} / W^{'} = 0$ in the direct sum $V / W^{'} = W / W^{'} \oplus V^{'} / W^{'}$ , so $W \cap L \subseteq W \cap V^{'} = W^{'}$ and $W \cap L \subseteq L$ give $W \cap L \subseteq W^{'} \cap L = 0$ ; and $\dim W + \dim L = (n - 1) + 1 = n = \dim V$ with $W + L \subseteq V$ , so $V = W \oplus L$ .

1.4F2givenalgebra

(b) $\Rightarrow$ (a), first the splitting property. Assume (b) and let $W \subseteq V$ be a subrepresentation of a finite-dimensional $V$ . If $W = 0$ , it is already a direct summand, so assume $W \neq 0$ . On the rational representation ${Hom}_{k} (V, W)$ of [F2] consider the subrepresentations $V_{1} = {f : f ∣_{W} = a {id}_{W} for some a \in k}$ and $W_{1} = {f : f ∣_{W} = 0}$ ; both are subrepresentations because $W$ is stable under $G$ , and $V_{1} / W_{1} ≅ k$ has dimension one. By (b) applied to the pair $W_{1} \subseteq V_{1}$ there is a one-dimensional subrepresentation $L$ with $V_{1} = W_{1} \oplus L$ . Every nonzero $f \in L$ satisfies $f ∣_{W} = a {id}_{W}$ with $a \neq 0$ ; by [F3] the one-dimensional representation $L$ is trivial, so $g \cdot f = f$ for all $R$ -points $g$ , that is, $f (v) = g \cdot f (g^{- 1} v)$ for all $v \in V$ , which after replacing $g$ by $g^{- 1}$ says $f (g \cdot v) = g \cdot f (v)$ : $f$ is a homomorphism of rational representations. Replacing $f$ by $a^{- 1} f$ we may suppose $f ∣_{W} = {id}_{W}$ , and then $V = W \oplus \ker f$ because $f ∣_{W} = {id}_{W}$ gives $W \cap \ker f = 0$ and $v - f (v) \in \ker f$ for every $v$ .

2.1F1step 1.4

(b) $\Rightarrow$ (a), conclusion. Under (b) every subrepresentation of every finite-dimensional $V$ is a direct summand by step 1.4. We prove by induction on $\dim V$ that such a $V$ is a direct sum of simple subrepresentations. For $V = 0$ this is the empty sum. If $V \neq 0$ , choose a nonzero subrepresentation $S \subseteq V$ of minimal dimension; it is simple, because a proper nonzero subrepresentation of $S$ would be a nonzero subrepresentation of $V$ of smaller dimension. By step 1.4, $V = S \oplus C$ with $\dim C < \dim V$ , and every subrepresentation of $C$ is a subrepresentation of $V$ , so the induction hypothesis applies to $C$ and exhibits it as a direct sum of simple subrepresentations; adjoining $S$ gives such a decomposition of $V$ .

3.1step 1.1step 1.2step 1.3step 1.4step 2.1∎

Steps 1.1, 1.2, 1.3, 1.4 and 2.1 prove (a) $\Rightarrow$ (b) $\Rightarrow$ (c) and (c) $\Rightarrow$ (b) $\Rightarrow$ (a), so the three conditions are equivalent.

Remarks

The hypothesis $X (G) = 0$ enters only in step 1.4, through the triviality of the one-dimensional representation $L$ ; it is what forces the constructed linear map $V \to W$ to be equivariant rather than merely $G$ -invariant as a line.
The proof is the source's proof of Lemma 22.40; the equivalence of the sum and direct-sum formulations of semisimplicity is not used, because the definition adopted here is the direct-sum one.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Semisimplicity of rational representations descends along field extensions

Statement

Let $G$ be an algebraic group over a field $k$ , let $(V, r)$ be a finite-dimensional rational representation and let $k^{'} \supseteq k$ be a field extension. If the base change $(V_{k^{'}}, r_{k^{'}})$ is semisimple as a representation of $G_{k^{'}}$ , then $(V, r)$ is semisimple (Rational representations and comodules of an affine group scheme, Simple and semisimple rational representations). In particular it suffices to test semisimplicity after extending scalars to an algebraic closure of $k$ . For a possibly nonaffine $G$ , rationality here means that $r : G \to {GL}_{V}$ is a morphism; subrepresentations are the invariant subspaces. This agrees with the cited comodule definition for affine $G$ .

Facts & Assumptions

Given: An algebraic group $G$ over $k$ , a finite-dimensional rational representation $(V, r)$ , a field extension $k^{'} \supseteq k$ , and the base changes $G_{k^{'}}$ and $(V_{k^{'}}, r_{k^{'}})$ . Put $A = Γ (G, O_{G})$ ; finite representation matrices have entries in $A$ , whether or not $G$ is affine.

[F1]

Base change of matrix coefficients. Base change of the morphism $r : G \to {GL}_{V}$ defines the representation on $V_{k^{'}}$ , and preserves invariant subspaces. A finite-dimensional subspace $C \subseteq A$ remains linearly independent after base change: the restriction maps from $C$ to the rings of affine opens jointly detect zero, and finitely many suffice. Indeed choose a finite intersection of their kernels of minimal dimension; if it were nonzero, another restriction would lower its dimension. Thus $C$ injects into a finite direct sum of affine-open coordinate rings. Tensoring with $k^{'}$ preserves this injection by finite coefficient comparison, and these rings become the coordinate rings of the base-changed affine opens by Affine fibre products are spectra of tensor products. Consequently $C \otimes_{k} k^{'} \to Γ (G_{k^{'}}, O_{G_{k^{'}}})$ is injective. In the affine case these are the usual comodule coefficient calculations of Rational representations and comodules of an affine group scheme; the argument does not require global affineness.

[F2]

Hom representation and scalar extension. For finite-dimensional $W$ , the space $M = {Hom}_{k} (V, W)$ with $(g \cdot f) (v) = g \cdot f (g^{- 1} v)$ is a finite-dimensional rational representation of $G$ (Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational). For nonaffine $G$ its matrices are still regular: they are the products of the representation matrix on $W$ and the inverse representation matrix on $V$ , so the displayed action gives a morphism into ${GL}_{M}$ directly. Choose a finite basis $(e_{i})$ of $V$ with dual basis $(e_{i}^{*})$ , and a finite basis $(w_{j})$ of $W$ . The rank-one maps $E_{i j} : v \mapsto e_{i}^{*} (v) w_{j}$ form a $k$ -basis of $M$ ; after scalar extension, their corresponding maps on $V_{k^{'}}$ with values in $W_{k^{'}}$ form a $k^{'}$ -basis of ${Hom}_{k^{'}} (V_{k^{'}}, W_{k^{'}})$ . Thus the canonical map $M \otimes_{k} k^{'} \to {Hom}_{k^{'}} (V_{k^{'}}, W_{k^{'}})$ sending $f \otimes a$ to $a f_{k^{'}}$ is an isomorphism (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ).

[F3]

Equivariance is a finite linear system. In a finite basis $m_{1}, \dots, m_{N}$ of $M$ , write $g \cdot m_{j} = \sum_{i} a_{i j} (g) m_{i}$ with $a_{i j} \in A$ . For $f = \sum_{j} z_{j} m_{j}$ , fixedness is exactly $\sum_{j} a_{i j} z_{j} = z_{i}$ as regular functions on $G$ , for every $i$ . Let $C \subseteq A$ be the span of $1$ and all $a_{i j}$ . Comparing coefficients in a finite basis of $C$ turns these identities into finitely many linear equations over $k$ . By [F1] that coefficient basis stays independent on $G_{k^{'}}$ , so the base-changed equations express exactly $G_{k^{'}}$ -fixedness. Fixedness in the Hom action means equivariance of $f$ . Adding the finite coordinate equations $f ∣_{W} = {id}_{W}$ defines the affine solution set $S = {f \in M : f is G -equivariant, f ∣_{W} = {id}_{W}} .$ This uses the finite matrix interpretation of rationality; in the affine case it is the usual comodule condition (Rational representations and comodules of an affine group scheme, [F2]).

[F4]

Linear systems over a field. Every finite matrix over a field is row equivalent to a matrix in reduced row echelon form, obtained by Gauss-Jordan elimination (Gauss–Jordan elimination reduces every finite matrix over a field to reduced row echelon form). Row operations are invertible and preserve solution sets over every extension field; a system in reduced row echelon form is solvable if and only if it has no row $(0 \dots 0 ∣ c)$ with $c \neq 0$ , and when it is solvable, setting the free variables equal to $0$ and solving the pivot equations gives a solution with coordinates in the field generated by the coefficients, hence in $k$ when the coefficients lie in $k$ .

[F5]

Splitting implies semisimplicity. If every subrepresentation of a finite-dimensional rational representation $U$ is a direct summand, then $U$ is semisimple: choose a nonzero subrepresentation of minimal dimension, which is simple, split it off, and iterate on the complement of smaller dimension (Simple and semisimple rational representations).

Proof

technique · direct

1.1F1givenalgebra

Assume that $(V_{k^{'}}, r_{k^{'}})$ is semisimple and let $W \subseteq V$ be a subrepresentation. Then $W_{k^{'}}$ is a subrepresentation of $V_{k^{'}}$ . Write $V_{k^{'}}$ as a finite direct sum of simple subrepresentations and choose a largest subfamily whose sum $C$ intersects $W_{k^{'}}$ trivially. If $W_{k^{'}} + C \neq V_{k^{'}}$ , some simple summand $S_{i}$ is not contained in $W_{k^{'}} + C$ ; simplicity gives $S_{i} \cap (W_{k^{'}} + C) = 0$ , so adjoining $S_{i}$ contradicts maximality. Hence $V_{k^{'}} = W_{k^{'}} \oplus C$ .

1.2F2F3

The set $S$ of $G$ -equivariant $k$ -linear maps $f : V \to W$ with $f ∣_{W} = {id}_{W}$ is, by [F3], the solution set of a finite system of linear equations with coefficients in $k$ , inside the finite-dimensional $k$ -vector space $M = {Hom}_{k} (V, W)$ ; and $M \otimes_{k} k^{'} ≅ {Hom}_{k^{'}} (V_{k^{'}}, W_{k^{'}})$ identifies the base-changed system with the corresponding system over $k^{'}$ .

2.1step 1.1step 1.2

The base-changed system has a solution over $k^{'}$ : the projection $p : V_{k^{'}} \to W_{k^{'}}$ along the decomposition $V_{k^{'}} = W_{k^{'}} \oplus C$ of step 1.1 is $G_{k^{'}}$ -equivariant and restricts to the identity on $W_{k^{'}}$ . Thus $p$ solves the equations of step 1.2 over $k^{'}$ ; the affine solution set $S$ itself need not be a vector space.

3.1F4step 1.2step 2.1

The system of step 1.2 has a solution over $k$ . Row-reduce its augmented matrix over $k$ by Gauss-Jordan elimination; the resulting reduced row echelon system has the same solution set over $k^{'}$ , so it is solvable over $k^{'}$ by step 2.1 and therefore has no row of the form $(0 \dots 0 ∣ c)$ with $c \neq 0$ ; setting the free variables equal to zero and solving the pivot equations then produces a solution in $k$ .

4.1step 3.1

Let $f \in S$ . Then $f : V \to W$ is $G$ -equivariant with $f ∣_{W} = {id}_{W}$ , so $W \cap \ker f = 0$ and every $v \in V$ satisfies $v - f (v) \in \ker f$ , giving $V = W \oplus \ker f$ ; thus every subrepresentation of $V$ is a direct summand.

5.1F5step 4.1

By [F5] the representation $(V, r)$ is semisimple.

6.1step 5.1∎

If in particular $V_{\overset{ˉ}{k}}$ is semisimple for an algebraic closure $\overset{ˉ}{k}$ of $k$ , applying step 5.1 to the extension $k \subseteq \overset{ˉ}{k}$ shows that $V$ is semisimple; this completes the proof.

Remarks

The extension $k^{'}$ need not be algebraic or separable: only the invariance of consistency of a $k$ -linear system under base change is used, which holds for every field extension.
The field extension enters twice: in defining the base-changed representation $V_{k^{'}}$ and in producing the $k^{'}$ -solution of the complement equations; the descent of the solution itself is elementary linear algebra over $k$ .

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The top exterior power detects stabilizers of a subspace

Statement

Let $V$ be a finite-dimensional vector space over a field $k$ , let $W \subseteq V$ be a subspace of dimension $d$ , and put $D = Λ^{d} W \subseteq Λ^{d} V$ (The $k$ th exterior power as the tensor-power quotient by repeated-vector relations, The induced map $Λ^{k} T$ on exterior powers, If $\dim V = n$ , then $\dim Λ^{k} V = (\binom{n}{k})$ ). Then for every $k$ -algebra $R$ and every $α \in GL (V_{R})$ one has $α W_{R} = W_{R}$ if and only if $(Λ^{d} α) (D_{R}) = D_{R}$ . In particular, if $(V, r)$ is a rational representation of an affine group scheme $G$ over $k$ and $G$ acts on $Λ^{d} V$ by the exterior power (On $Λ^{n} V$ , the induced map $Λ^{n} T$ is multiplication by $\det T$ ), then the scheme-theoretic stabilizer of $W$ in $G$ equals the scheme-theoretic stabilizer of the line $D$ .

Facts & Assumptions

Given: A field $k$ , a finite-dimensional $k$ -vector space $V$ , a subspace $W \subseteq V$ of dimension $d \geq 1$ , a $k$ -basis $e_{1}, \dots, e_{d}$ of $W$ extended to a $k$ -basis $e_{1}, \dots, e_{n}$ of $V$ , and the element $w = e_{1} \land \dots \land e_{d} \in Λ^{d} V$ . For a $k$ -algebra $R$ we write $V_{R} = V \otimes_{k} R$ with its $R$ -basis $e_{1}, \dots, e_{n}$ , $W_{R} = R e_{1} + \dots + R e_{d}$ , and $D_{R} = R w \subseteq Λ_{R}^{d} (V_{R})$ , where $Λ_{R} (V_{R})$ is the exterior algebra of the finite free $R$ -module $V_{R}$ (Exterior Algebra Of A Finite Free Module).

[F1]

Wedge basis over a field. For an ordered basis $(e_{1}, \dots, e_{n})$ of a finite-dimensional vector space and $0 \leq k \leq n$ , the increasing wedges $e_{I} = e_{i_{1}} \land \dots \land e_{i_{k}}$ indexed by the $k$ -element subsets $I \subseteq {1, \dots, n}$ form a basis of $Λ^{k} V$ (Increasing-index wedges of a basis form a basis of $Λ^{k} V$ ). In particular $Λ^{d} V$ has the basis vector $w$ indexed by ${1, \dots, d}$ , and $\dim Λ^{k} V = (\binom{n}{k})$ (If $\dim V = n$ , then $\dim Λ^{k} V = (\binom{n}{k})$ ).

[F2]

Exterior algebra of a finite free module. For a commutative unital ring $R$ and a finite free $R$ -module $F$ with ordered basis $f_{1}, \dots, f_{n}$ , the exterior algebra is the graded quotient $⋀_{R} F = T_{R} (F) / (v \otimes v : v \in F)$ of the tensor algebra by the two-sided ideal generated by the elements $v \otimes v$ , and $v_{1} \land \dots \land v_{m}$ denotes the image of $v_{1} \otimes \dots \otimes v_{m}$ ; the pure tensors of basis elements form an $R$ -basis of $T_{R} (F)$ (Exterior Algebra Of A Finite Free Module).

[F3]

Leibniz determinant over a commutative ring. For every commutative ring $R$ and $m \geq 1$ the Leibniz determinant $\det : M_{m} (R) \to R$ of For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix is multilinear in the columns, alternating, and normalized, $\det I_{m} = 1$ (The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring); in particular $\det (c_{1}, \dots, c_{m}) = 0$ as soon as one column is the zero column, because column multilinearity factors out the scalar $0$ .

[F4]

Scheme-theoretic stabilizers. The scheme-theoretic stabilizer $G_{x}$ of a $k$ -point $x$ of a finite-type $k$ -scheme $X$ with $G$ -action is the closed subgroup scheme of $G$ whose $R$ -points are $G_{x} (R) = {g \in G (R) : g \cdot x_{R} = x_{R}}$ for every $k$ -algebra $R$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers); a closed subscheme of an affine scheme is determined by its functor of points.

[F5]

Functoriality of exterior powers over $k$ . A linear map of $k$ -vector spaces induces $Λ^{k} T$ with $Λ^{k} T (v_{1} \land \dots \land v_{k}) = T v_{1} \land \dots \land T v_{k}$ , and $Λ^{k} (id) = id$ with $Λ^{k} (S \circ T) = Λ^{k} S \circ Λ^{k} T$ (The induced map $Λ^{k} T$ on exterior powers, Exterior powers represent alternating multilinear maps and are unique up to unique isomorphism); for an endomorphism $T$ of an $n$ -dimensional space, $Λ^{n} T = \det (T) id$ (On $Λ^{n} V$ , the induced map $Λ^{n} T$ is multiplication by $\det T$ ).

[F6]

Exterior powers of rational representations. For a finite-dimensional rational representation $(V, r)$ , the exterior power $Λ^{d} V$ with $g \cdot (v_{1} \land \dots \land v_{d}) = g \cdot v_{1} \land \dots \land g \cdot v_{d}$ is a rational representation, and for every $k$ -algebra $R$ the induced action on $Λ_{R}^{d} (V_{R})$ is $Λ_{R}^{d} (r_{R} (g))$ (Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational).

Proof

technique · direct

1.1F1F2given

Let $R$ be a $k$ -algebra. The basis $e_{1}, \dots, e_{n}$ of $V$ is an $R$ -basis of $V_{R}$ , and $W_{R}$ is the free direct summand $R e_{1} \oplus \dots \oplus R e_{d}$ because $W$ is spanned by $e_{1}, \dots, e_{d}$ and is complemented by the span of $e_{d + 1}, \dots, e_{n}$ . We work in the graded exterior algebra $⋀_{R} (V_{R})$ of [F2] and write $Λ_{R}^{k} (V_{R})$ for its degree- $k$ part; for $k \leq n$ this is a free $R$ -module, and the field case $R = k$ agrees with the exterior power of the statement by [F1], both having the wedges $e_{I}$ as basis.

1.2F2F5

Functoriality on $⋀_{R} (V_{R})$ . An $R$ -linear map $α : V_{R} \to V_{R}$ induces the algebra map $T (α)$ of the tensor algebra with $α (v_{1} \otimes \dots \otimes v_{m}) = α v_{1} \otimes \dots \otimes α v_{m}$ ; it maps the defining ideal into itself because $v \otimes v \mapsto α v \otimes α v$ and the ideal is generated by these elements, so it descends to a graded $R$ -linear map $Λ (α)$ with $Λ (α) (v_{1} \land \dots \land v_{m}) = α v_{1} \land \dots \land α v_{m}$ , equal to $Λ_{R}^{m} (α)$ in degree $m$ . If $α$ is invertible with inverse $β$ , then $Λ (α) Λ (β) = Λ (β) Λ (α) = id$ on wedges and hence on the whole algebra by linearity, so each $Λ_{R}^{m} (α)$ is an $R$ -linear automorphism; and for an $R$ -submodule $U \subseteq V_{R}$ one has $Λ_{R}^{m} (α) (Λ_{R}^{m} U) = Λ_{R}^{m} (α U)$ , where $Λ_{R}^{m} U$ is the span of $m$ -fold wedges of elements of $U$ . For $R = k$ this is [F5].

2.1F2step 1.1

The increasing wedges span $Λ_{R}^{k} (V_{R})$ . For $k \geq 1$ every element of $⋀_{R} (V_{R})$ is an $R$ -linear combination of wedges of basis vectors, because $T_{R} (V_{R})$ is spanned by pure tensors of basis vectors ([F2]) and the wedge is the quotient map. In the quotient, $u \land u = 0$ for every $u \in V_{R}$ , since $u \otimes u$ lies in the ideal of [F2]; applying this to $u + w$ and expanding bilinearly gives $u \land w = - w \land u$ for all $u, w$ , so adjacent transpositions change a wedge by a sign. Consequently any wedge of basis vectors with two equal entries is zero: move the equal entries next to each other by adjacent transpositions, the wedge picks up a sign, and the adjacent product $u \land u$ vanishes. Every wedge of basis vectors with pairwise distinct indices therefore equals $\pm e_{I}$ for the corresponding increasing subset $I$ , and the $e_{I}$ span $Λ_{R}^{k} (V_{R})$ ; for $k = 0$ the element $1$ spans $Λ_{R}^{0} (V_{R}) = R$ .

2.2F2F3step 1.1

The increasing wedges are $R$ -independent. Fix a $k$ -element subset $J = {j_{1} < \dots < j_{k}} \subseteq {1, \dots, n}$ and define, for $v_{1}, \dots, v_{k} \in V_{R}$ , the element $φ_{J} (v_{1}, \dots, v_{k}) \in R$ as the Leibniz determinant of the $k \times k$ matrix whose entry in row $m$ and column $ℓ$ is the coefficient of $e_{j_{m}}$ in $v_{ℓ}$ ; its columns depend $R$ -linearly on $v_{1}, \dots, v_{k}$ and equal columns occur when two of the vectors are equal, so by [F3] it is $R$ -multilinear and alternating in $v_{1}, \dots, v_{k}$ . Extend $φ_{J}$ $R$ -linearly to a map ${\tilde{ψ}}_{J} : T_{R} (V_{R}) \to R$ on the basis of pure tensors of basis vectors of [F2], declaring it zero in tensor degree different from $k$ . Then ${\tilde{ψ}}_{J}$ vanishes on every element $t (v \otimes v) t^{'}$ with $t, t^{'}$ pure tensors of basis vectors: expanding $v = \sum_{r} a_{r} e_{r}$ , such an element is a combination of pure tensors with two adjacent entries $e_{r}, e_{s}$ in the middle, and in degree $k$ it evaluates to $\sum_{r, s} a_{r} a_{s} φ_{J} (\dots, e_{r}, e_{s}, \dots)$ , which is zero because a bilinear form alternating in two adjacent arguments satisfies $\sum_{r, s} a_{r} a_{s} β (e_{r}, e_{s}) = 0$ over any commutative ring (the sum has $\sum_{r < s} a_{r} a_{s} (β (e_{r}, e_{s}) + β (e_{s}, e_{r})) + \sum_{r} a_{r}^{2} β (e_{r}, e_{r}) = 0$ ). As these elements $R$ -span the defining ideal of [F2], ${\tilde{ψ}}_{J}$ descends to an $R$ -linear map $ψ_{J} : ⋀_{R} (V_{R}) \to R$ . The matrix whose columns are $e_{i_{1}}, \dots, e_{i_{k}}$ in the rows $J$ is the identity matrix indexed by $J$ when $I = {i_{1} < \dots < i_{k}} = J$ , so $ψ_{J} (e_{J}) = 1$ by [F3]; and when $I \neq J$ some $ℓ$ has $i_{ℓ} \notin J$ (as $∣ I ∣ = ∣ J ∣$ ), so column $ℓ$ is a zero column and $ψ_{J} (e_{I}) = 0$ by [F3]. Therefore $\sum_{I} c_{I} e_{I} = 0$ forces $c_{J} = ψ_{J} (0) = 0$ for every $J$ , and the $e_{I}$ are $R$ -independent.

3.1step 2.1step 2.2

Annihilator description of $W_{R}$ . An element of $V_{R}$ is uniquely $v = \sum_{i = 1}^{n} a_{i} e_{i}$ with $a_{i} \in R$ , and $w \land v = \sum_{i = 1}^{n} a_{i} e_{1} \land \dots \land e_{d} \land e_{i} = \sum_{i = d + 1}^{n} a_{i} e_{{1, \dots, d, i}}$ , since wedges with a repeated index vanish by step 2.1. By step 2.2 the family of wedges $e_{{1, \dots, d, i}}$ , $d + 1 \leq i \leq n$ , is part of the $R$ -basis of $Λ_{R}^{d + 1} (V_{R})$ and so is $R$ -independent; hence $w \land v = 0$ if and only if $a_{i} = 0$ for all $i > d$ , that is, if and only if $v \in W_{R}$ .

3.2step 2.1step 1.2

If $α W_{R} = W_{R}$ then $(Λ^{d} α) (D_{R}) = D_{R}$ . The submodule $Λ_{R}^{d} (W_{R})$ of $Λ_{R}^{d} (V_{R})$ is generated by $w$ : any $d$ -fold wedge of elements of $W_{R}$ expands in the $R$ -basis $e_{1}, \dots, e_{d}$ of $W_{R}$ into a combination of wedges with repeated indices and of the single wedge $w$ . Hence $D_{R} = Λ_{R}^{d} (W_{R})$ , and step 1.2 gives $(Λ_{R}^{d} α) (D_{R}) = Λ_{R}^{d} (α W_{R}) = Λ_{R}^{d} (W_{R}) = D_{R}$ .

4.1step 1.2step 3.1

If $(Λ^{d} α) (D_{R}) = D_{R}$ then $α W_{R} = W_{R}$ . Since $D_{R} = R w$ and $Λ_{R}^{d} α$ is an automorphism of $Λ_{R}^{d} (V_{R})$ preserving $D_{R}$ (step 1.2), its restriction $D_{R} \to D_{R}$ is bijective and therefore $Λ_{R}^{d} α (w) = c w$ for a unique unit $c \in R^{\times}$ . Let $v \in W_{R}$ ; by step 3.1 $w \land v = 0$ , so $0 = Λ_{R}^{d + 1} α (w \land v) = (Λ_{R}^{d} α) (w) \land Λ_{R} α (v) = c (w \land α v)$ , and since $c$ is a unit, $w \land α v = 0$ , so $α v \in W_{R}$ by step 3.1. Thus $α W_{R} \subseteq W_{R}$ . The same argument applied to $α^{- 1}$ , whose induced map $(Λ_{R}^{d} α)^{- 1}$ also preserves $D_{R}$ , gives $α^{- 1} W_{R} \subseteq W_{R}$ , hence $W_{R} \subseteq α W_{R}$ , and therefore $α W_{R} = W_{R}$ .

5.1F6step 3.2step 4.1

The stabilizers agree. Let $(V, r)$ be a rational representation of an affine group scheme $G$ over $k$ and let $g \in G (R)$ for a $k$ -algebra $R$ ; the $R$ -point of the representation is the $R$ -linear automorphism $r_{R} (g)$ of $V_{R}$ , and by [F6] the induced action on $Λ^{d} V$ is the rational action $Λ_{R}^{d} (r_{R} (g))$ . Steps 3.2 and 4.1 give $r_{R} (g) W_{R} = W_{R}$ if and only if $(Λ_{R}^{d} (r_{R} (g))) (D_{R}) = D_{R}$ , that is, the $R$ -points of the stabilizer of the subspace $W$ and of the stabilizer of the line $D$ coincide for every $k$ -algebra $R$ .

6.1F4step 5.1∎

By [F4] the scheme-theoretic stabilizers of $W$ and of the line $D$ are the closed subgroup schemes of $G$ defined by exactly these two functors of points; a closed subscheme of an affine scheme is determined by its functor of points, so the two closed subgroup schemes are equal. This completes both the base-change equivalence for every $k$ -algebra $R$ and the scheme-theoretic stabilizer statement.

Remarks

The hypothesis $d \geq 1$ is the only nondegenerate case: for $d = 0$ the space $W = 0$ has $D = Λ^{0} W = k$ and the statement is also true, as both stabilizers are all of $G$ ; the proof above covers $d \geq 1$ and the degenerate case is immediate.
The equivalence is proved over every $k$ -algebra $R$ , not only over fields, because that is exactly what the scheme-theoretic stabilizer statement requires: it is the $R$ -points for all $R$ that determine the closed subgroup scheme.

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Lie ideals and normal connected subgroups in characteristic zero

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be a smooth connected affine algebraic group over a field $k$ of characteristic $0$ , let $H \subseteq G$ be a smooth closed subgroup scheme with identity component $H^{\circ}$ , and let $h = Lie (H) \subseteq g = Lie (G)$ (The adjoint representation of an affine group scheme, The Lie bracket from infinitesimals and the adjoint action). Then: (a) $[g, h] \subseteq h$ if and only if $H^{\circ}$ is normal in $G$ ; (b) if $H$ is normal in $G$ , then $[g, h] \subseteq h$ . In particular, if $H$ is connected, then $H$ is normal in $G$ if and only if $h$ is an ideal of $g$ .

Facts & Assumptions

Given: A smooth connected affine group scheme $G$ over a characteristic-zero field $k$ with $g = Lie (G)$ , a smooth closed subgroup scheme $H \subseteq G$ with $h = Lie (H)$ , and the adjoint representation $Ad : G \to {GL}_{g}$ .

[F1]

Adjoint action and its differential. For every $R$ -point $x \in G (R)$ the conjugation automorphism $c_{x} : G_{R} \to G_{R}$ satisfies $Lie (c_{x}) = Ad (x)$ on $g_{R}$ , and $x e^{ε X} x^{- 1} = e^{ε Ad (x) X}$ ; the differential $ad = Lie (Ad)$ is the bracket, $[X, Y] = ad (X) Y$ , so that $Ad (e^{ε Y}) = id + ε ad (Y)$ on $g \otimes k [ε]$ (The adjoint representation of an affine group scheme, The Lie bracket from infinitesimals and the adjoint action).

[F2]

Lie-stable subspaces are stable. If $k$ has characteristic $0$ , $G$ is connected and smooth, $(V, r)$ is a rational representation and $W \subseteq V$ satisfies $g W \subseteq W$ , then $W$ is $G$ -stable (Lie algebras of subspace stabilizers and Lie-stable subspaces).

[F3]

Connected subgroups with equal Lie algebras. If $H_{1} \subseteq H_{2}$ are closed subgroup schemes of a connected algebraic group, $H_{1}$ and $H_{2}$ are smooth, $H_{2}$ is connected and $Lie (H_{1}) = Lie (H_{2})$ , then $H_{1} = H_{2}$ (The Lie functor: exactness, fixed points and generation).

The same supplier, part (c), identifies $Lie (N_{G} (H^{\circ}))$ with the inverse image of $(g / h)^{H^{\circ}}$ in $g$ ; this is a statement about the normalizer Lie algebra, not the Lie algebra of a quotient group. The Lie functor: exactness, fixed points and generation

[F4]

Cartier. Every affine group scheme of finite type over a characteristic-zero field is smooth (Cartier's theorem: affine group schemes in characteristic zero are smooth; AC is used here and is inherited by this item).

Proof

technique · direct

1.1F1given

For every $k$ -algebra $R$ and $x \in G (R)$ , $Lie (c_{x}) = Ad (x)$ on $g_{R}$ , and for $Y \in g$ the dual-number point $e^{ε Y} \in G (k [ε])$ acts by $Ad (e^{ε Y}) = id + ε ad (Y)$ on $g \otimes_{k} k [ε]$ .

1.2F4given

The identity component. $H^{\circ}$ is a smooth closed connected subgroup scheme of $G$ with $Lie (H^{\circ}) = h$ : it is smooth by Cartier's theorem, and the identity component of the smooth group scheme $H$ has the same Lie algebra as $H$ .

2.1F1step 1.1

Part (b). Assume that $H$ is normal in $G$ . For $Y \in g$ the point $e^{ε Y} \in G (k [ε])$ normalizes $H_{k [ε]}$ , so by step 1.1 the automorphism $Lie (c_{e^{ε Y}}) = id + ε ad (Y)$ of $g \otimes k [ε]$ preserves $h \otimes k [ε]$ . Writing $X \in h$ as $X$ , this says $X + ε [Y, X] \in h \otimes k [ε]$ , hence $[Y, X] \in h$ ; as $Y \in g$ was arbitrary, $[g, h] \subseteq h$ .

2.2F2F3F4step 1.1step 1.2algebra

Part (a), reverse. Assume $[g, h] \subseteq h$ . By [F2] applied to $(g, Ad)$ , the subspace $h$ is $G$ -stable. Put $N = N_{G} (H^{\circ})$ , a closed affine subgroup scheme of $G$ . The normalizer formula in The Lie functor: exactness, fixed points and generation, part (c), identifies $Lie (N)$ with the inverse image of $(g / h)^{H^{\circ}}$ . Since $h$ is a Lie ideal, the differential action of $Lie (H^{\circ}) = h$ on $Q = g / h$ is zero. Equip $Q \oplus k$ with the quotient representation and a trivial last summand. For each $v \in Q$ , the line $k (v, 1)$ is Lie-stable, hence $H^{\circ}$ -stable by [F2] applied to the smooth connected characteristic-zero group $H^{\circ}$ . For every $R$ and $h \in H^{\circ} (R)$ , its last coordinate forces the scalar by which $h$ preserves this line to be $1$ , so $h (v, 1) = (v, 1)$ . Thus every $v$ is fixed and the quotient representation is trivial. Consequently $(g / h)^{H^{\circ}} = g / h$ , and $Lie (N) = g$ . Cartier [F4] makes $N$ smooth; [F3] for the nested inclusion $N \subseteq G$ now gives $N = G$ . Therefore $H^{\circ}$ is normal in $G$ .

3.1step 2.1step 1.2

Part (a), forward. If $H^{\circ}$ is normal in $G$ , then step 2.1 applied to the normal smooth closed subgroup $H^{\circ}$ with Lie algebra $h$ gives $[g, h] \subseteq h$ .

4.1step 2.1step 3.1step 2.2∎

If $H$ is connected then $H = H^{\circ}$ and steps 3.1 and 2.2 give the equivalence of normality with $h$ being an ideal of $g$ ; steps 2.1, 3.1 and 2.2 together prove all three assertions.

Remarks

The connectedness of $H$ is essential in the equivalence: a finite non-normal subgroup $H$ of a connected group in characteristic $0$ has $h = 0$ , so $[g, h] \subseteq h$ holds while $H$ is not normal (for instance a subgroup of order $2$ in ${PGL}_{2}$ ). Only $H^{\circ}$ is detected by the Lie algebra, and the statement is worded accordingly.
The hypothesis that $k$ has characteristic $0$ enters through Cartier's theorem and through the Lie-stable-subspace lemma; both are used to pass from the infinitesimal condition to the group.

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Chevalley: every closed subgroup is a line stabilizer

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be an affine group scheme of finite type over a field $k$ and let $H \subseteq G$ be a closed subgroup scheme. Then there exist a finite-dimensional rational representation $(V, r)$ of $G$ and a line $L \subseteq V$ such that $H = {Stab}_{G} (L)$ scheme-theoretically: for every $k$ -algebra $R$ , ${Stab}_{G} (L) (R) = {g \in G (R) : g L_{R} = L_{R}}$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers, Rational representations and comodules of an affine group scheme). Moreover, if $L = k v$ then $Lie (H) = {x \in Lie (G) : x v \in k v}$ .

Facts & Assumptions

Given: An affine group scheme $G$ of finite type over $k$ with coordinate Hopf algebra $A = O (G)$ and a closed subgroup scheme $H \subseteq G$ , with kernel $a = \ker (A \to O (H))$ .

[F1]

Hopf ideals and closed subgroups. $a$ is a Hopf ideal of $A$ and $H = Spec (A / a)$ ; in particular $Δ (a) \subseteq A \otimes a + a \otimes A$ and $ε (a) = 0$ (Closed subgroup schemes of an affine group scheme correspond to Hopf ideals, Hopf ideals, kernels and quotients of commutative Hopf algebras, The coordinate Hopf algebra of an affine group scheme).

[F2]

Finiteness. $A$ is a finitely generated $k$ -algebra (An affine scheme of finite type over a field has a finitely generated coordinate ring) and finitely generated algebras over the Noetherian field $k$ are Noetherian, so $a$ is finitely generated as an ideal (Every algebra of finite type over a Noetherian ring is a Noetherian ring).

[F3]

Finite-dimensional subcomodules. $A$ is a comodule over itself under $Δ$ , and every finite subset of a comodule lies in a finite-dimensional subcomodule and subrepresentations are subcomodules (Rational representations of an affine group scheme are comodules of its coordinate Hopf algebra, Every element of a comodule lies in a finite-dimensional subcomodule, Rational representations and comodules of an affine group scheme).

[F4]

Exterior-power stabilizers. For a finite-dimensional rational representation $(V, r)$ and a subspace $W \subseteq V$ of dimension $d$ , the scheme-theoretic stabilizer of $W$ equals the scheme-theoretic stabilizer of the line $Λ^{d} W \subseteq Λ^{d} V$ , and $Λ^{d} V$ with the exterior-power action is a rational representation (The top exterior power detects stabilizers of a subspace, Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational).

[F5]

Lie algebras of stabilizers. For a subspace $W$ of a rational representation, $Lie ({Stab}_{G} (W)) = {x \in Lie (G) : x W \subseteq W}$ (Lie algebras of subspace stabilizers and Lie-stable subspaces).

Proof

technique · direct

1.1F1F2F3given

Since $a$ is finitely generated as an ideal by [F2], choose a finite generating set $S \subseteq a$ . By [F3] there is a finite-dimensional subcomodule $V \subseteq A$ under $Δ$ with $S \subseteq V$ ; put $W = a \cap V$ , choose a basis $(e_{j})_{j \in J}$ of $W$ and extend it to a basis $(e_{i})_{i \in J \cup I}$ of $V$ , so that $I$ indexes a complement of $W$ in $V$ . Write $Δ (e_{j}) = \sum_{i \in J \cup I} e_{i} \otimes a_{i j}$ for $j \in J$ and let $a^{'}$ be the ideal of $A$ generated by the elements $a_{i j}$ with $j \in J$ , $i \in I$ .

2.1F1step 1.1given

For a $k$ -algebra $R$ and $g \in G (R)$ one has $g \cdot e_{j} = \sum_{i \in J \cup I} e_{i} a_{i j} (g)$ under the action associated with the coaction $Δ$ , and the $e_{i}$ form an $R$ -basis of $V_{R}$ ; hence $g \cdot W_{R} \subseteq W_{R}$ if and only if $a_{i j} (g) = 0$ for all $j \in J$ , $i \in I$ , that is, if and only if $g$ vanishes on $a^{'}$ . Since $g$ acts invertibly and $W_{R}$ is a direct summand of $V_{R}$ , the inclusion $g \cdot W_{R} \subseteq W_{R}$ is equivalent to $g \cdot W_{R} = W_{R}$ . Therefore the stabilizer functor of $W$ is represented by the closed subscheme $Spec (A / a^{'})$ of $G = Spec A$ .

2.2F1step 1.1

$a^{'} \subseteq a$ : as $a$ is a Hopf ideal, $Δ (e_{j}) \in A \otimes a + a \otimes A$ for $j \in J$ by [F1], so applying $id \otimes π$ for the quotient $π : A \to A / a$ and then $q \otimes id$ for the quotient $q : A \to A / a$ gives $\sum_{i \in I} q (e_{i}) \otimes π (a_{i j}) = 0$ in $(A / a) \otimes (A / a)$ , because $π (e_{j}) = 0$ for $j \in J$ and $a \cap V = W$ ; the elements $q (e_{i})$ , $i \in I$ , are linearly independent, so $π (a_{i j}) = 0$ , that is, $a_{i j} \in a$ for all $j \in J$ , $i \in I$ .

2.3F1step 1.1

$a \subseteq a^{'}$ : for $j \in J$ one has $ε (e_{j}) = 0$ by [F1], and the counit axiom gives $e_{j} = (ε \otimes id) Δ (e_{j}) = \sum_{i \in J \cup I} ε (e_{i}) a_{i j} = \sum_{i \in I} ε (e_{i}) a_{i j} \in a^{'}$ , because the terms with $i \in J$ have $ε (e_{i}) = 0$ . The elements $e_{j}$ , $j \in J$ , span $W \supseteq S$ , and $S$ generates $a$ as an ideal, so $a$ is contained in the ideal $a^{'}$ .

3.1F1step 2.1step 2.2step 2.3

By steps 2.2 and 2.3, $a^{'} = a$ , so $Spec (A / a^{'}) = Spec (A / a) = H$ . Step 2.1 identifies $Spec (A / a^{'})$ with the stabilizer of $W$ , so $H = {Stab}_{G} (W)$ scheme-theoretically, where $V$ is a finite-dimensional rational representation of $G$ and $W \subseteq V$ .

4.1F4step 2.1

Let $d = \dim W$ and $L = Λ^{d} W \subseteq Λ^{d} V$ ; this is a line, $Λ^{d} V$ is a finite-dimensional rational representation of $G$ by [F4], and [F4] gives ${Stab}_{G} (W) = {Stab}_{G} (L)$ scheme-theoretically. Combined with step 3.1 this produces the required pair $(V, L)$ with $H = {Stab}_{G} (L)$ .

5.1F5step 4.1

If $L = k v$ , then ${x \in Lie (G) : x L \subseteq L} = {x : x v \in k v}$ and [F5] applied to the line $L$ gives $Lie (H) = Lie ({Stab}_{G} (L)) = {x \in Lie (G) : x v \in k v}$ .

6.1step 4.1step 5.1∎

Steps 4.1 and 5.1 prove both assertions of the theorem for the closed subgroup scheme $H$ of $G$ .

Remarks

The construction is Milne's proof of Theorem 4.27: the ideal $a$ is replaced by the ideal of matrix coefficients $a^{'}$ cut out by the finite-dimensional subcomodule $V$ , and the computation $a^{'} = a$ identifies $H$ with the stabilizer of $W = a \cap V$ in the regular representation restricted to $V$ .
The passage from the subspace $W$ to the line $L = Λ^{d} W$ is Lemma 4.28, which is where the exterior power of a rational representation and the scheme-theoretic stabilizer comparison are used.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

The Lie algebra of a semisimple group in characteristic zero is semisimple

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $0$ (Split reductive groups, Radical, unipotent radical, semisimple and reductive algebraic groups). Then its Lie algebra $g = Lie (G)$ is semisimple: it has no nonzero solvable ideal.

Facts & Assumptions

Given: A semisimple algebraic group $G$ over a characteristic-zero field $k$ , with $g = Lie (G)$ and the adjoint representation $Ad : G \to {GL}_{g}$ (The adjoint representation of an affine group scheme).

[F1]

The derived series of a Lie algebra is defined by successive brackets; Jacobi makes the successive derived terms of an ideal ideals in the ambient algebra. (Derived series and solvable Lie algebras)

[F2]

Stabilizers and their Lie algebras. For a finite-dimensional rational representation $(V, r)$ and a $k$ -point $w \in V (k)$ , the stabilizer of the point $w$ is a closed subgroup scheme of $G$ with $Lie ({Stab}_{G} (w)) = {x \in g : x w = 0}$ ; for the adjoint representation this is the computation $Ad (e^{ε x}) = id + ε ad (x)$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers, Fibres of the orbit map and the scheme-theoretic stabilizer as a closed subgroup scheme, The adjoint representation of an affine group scheme).

[F3]

Normal subgroups and ideals. For a smooth closed subgroup scheme $H \subseteq G$ , the identity component $H^{\circ}$ is normal in $G$ if and only if $Lie (H)$ is an ideal of $g$ (Lie ideals and normal connected subgroups in characteristic zero).

[F4]

Cartier's theorem makes every affine finite-type group scheme in characteristic $0$ smooth. The Lie algebra of a scheme-theoretic centralizer is the fixed subspace under the adjoint action; the differential of the adjoint action is the bracket. (The Lie functor: exactness, fixed points and generation, The adjoint representation of an affine group scheme, Cartier's theorem: affine group schemes in characteristic zero are smooth)

[F5]

The radical of a semisimple group. $G$ is semisimple exactly when $R (G) = 1$ , and a semisimple group is reductive; the radical $R (G)$ is the largest smooth connected normal solvable subgroup (Centre, radical and semisimple quotient of a reductive group, Radical, unipotent radical, semisimple and reductive algebraic groups).

Proof

Proof technique: direct.

1.1F1givenalgebra

If a nonzero solvable ideal $r$ exists, take its last nonzero derived term. Jacobi gives $[g, [I, I]] \subseteq [I, I]$ for each ambient ideal $I$ , so this term is an ambient ideal, and its next derived term being zero makes it commutative. Thus it suffices to exclude nonzero commutative ideals.

1.2givenalgebra

Let $n \subseteq g$ be a commutative ideal and put $h = {x \in g : [x, n] = 0}$ . Then $n \subseteq h$ because $n$ is commutative, and $h$ is an ideal of $g$ : for $y \in g$ , $x \in h$ and $n \in n$ one has $[[y, x], n] = [y, [x, n]] - [x, [y, n]] = 0$ because $[x, n] = 0$ and $[y, n] \in n$ is killed by $x$ .

2.1F2step 1.2

Choose a $k$ -basis $y_{1}, \dots, y_{m}$ of $n$ and let $H$ be the stabilizer of the point $(y_{1}, \dots, y_{m})$ of the rational representation $g^{\oplus m}$ ; this is a closed subgroup scheme of $G$ , and $Lie (H) = h$ : a dual-number point $e^{ε x}$ lies in $H$ exactly when $Ad (e^{ε x}) y_{i} = y_{i}$ for all $i$ , that is, by [F2], exactly when $[x, y_{i}] = 0$ for all $i$ .

3.1F3F4step 1.2step 2.1

Cartier's theorem [F4] makes the closed affine group $H$ smooth. Since $h = Lie (H)$ is an ideal of $g$ by step 1.2, [F3] shows that $H^{\circ}$ is normal in $G$ .

4.1F4step 3.1algebra

For any smooth connected affine $H_{0}$ in characteristic $0$ , $Z (H_{0}) = \ker {Ad}_{H_{0}}$ . Indeed, over an algebraic closure a point of the latter has centralizer with full Lie algebra by [F4]. Cartier makes that centralizer smooth, so it has full dimension and equals connected $H_{0}$ . Hence the geometric points of the adjoint kernel and centre agree; Cartier makes both subgroup schemes smooth and reduced, so they agree as schemes, and the equality descends to $k$ . Differentiating the adjoint kernel now gives $Lie Z (H_{0}) = \ker ad = Z (Lie H_{0})$ . Apply this to $H_{0} = H^{\circ}$ : its centre is characteristic in $H^{\circ}$ , hence normal in $G$ by step 3.1, and its Lie algebra is $Z (h)$ .

5.1F4F5step 4.1

$Z (H^{\circ})$ is finite: its identity component is a connected commutative, hence solvable, normal subgroup of $G$ , so it is contained in $R (G) = 1$ by [F5]; since $Z (H^{\circ})$ is smooth of dimension $0$ in characteristic $0$ , its Lie algebra is zero.

6.1step 1.1step 1.2step 4.1step 5.1∎

Finally $n \subseteq Z (h)$ : for $n \in n \subseteq h$ and $x \in h$ one has $[x, n] = 0$ by the definition of $h$ . Hence $n \subseteq Z (h) = Lie (Z (H^{\circ})) = 0$ , so $n = 0$ , and by step 1.1 the algebra $g$ has no nonzero solvable ideal.

Remarks

The proof is Milne's argument in the paragraph before Lemma 22.39: a commutative ideal $n$ has a centralizer $h$ that is again an ideal. The identity component of $Z (H^{\circ})$ is a normal connected commutative subgroup of $G$ , hence trivial; the full centre is finite, so its Lie algebra is zero in characteristic $0$ .
Characteristic $0$ is used twice: Cartier's theorem for smoothness of $H^{\circ}$ , $Z (H^{\circ})$ and $G$ , and the Lie-normal-subgroup correspondence.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Semisimple groups are perfect and have no nontrivial characters

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be a semisimple algebraic group over a field $k$ (Split reductive groups, Radical, unipotent radical, semisimple and reductive algebraic groups). Then $G = [G, G]$ (The derived subgroup, the derived series and solvable algebraic groups) and $X (G) = {Hom}_{k} (G, G_{m}) = 0$ ; equivalently, every one-dimensional rational representation of $G$ is trivial.

Facts & Assumptions

Given: A semisimple algebraic group $G$ over $k$ , its derived subgroup $[G, G] = G^{'}$ and its character group $X (G) = {Hom}_{k} (G, G_{m})$ .

[F1]

Semisimple groups are reductive. $R_{u} (G)$ is contained in the radical $R (G)$ and $G$ is semisimple exactly when $R (G_{k^{a}}) = 1$ , so $R_{u} (G_{k^{a}}) = 1$ and $G$ is reductive (Radical, unipotent radical, semisimple and reductive algebraic groups).

[F2]

Centre, radical and derived subgroup. For a reductive $G$ one has $G = Z (G)_{t} \cdot G^{'}$ with finite intersection, and $G^{'}$ is semisimple; moreover $G$ is semisimple if and only if $Z (G)$ is finite (Centre, radical and semisimple quotient of a reductive group, The derived subgroup, the derived series and solvable algebraic groups).

[F3]

Characters kill commutators. A morphism of algebraic groups $χ : G \to G_{m}$ satisfies $χ (g h g^{- 1} h^{- 1}) = 1$ for all $R$ -points $g, h$ , since $G_{m}$ is commutative; hence $χ$ is trivial on the derived subgroup $[G, G]$ (Properties of the derived subgroup of an algebraic group).

[F4]

One-dimensional representations are characters. A rational representation of $G$ on a one-dimensional $k$ -space is given by a morphism $G \to {GL}_{1} ≅ G_{m}$ , that is, by an element of $X (G)$ (Rational representations and comodules of an affine group scheme).

Proof

technique · direct

1.1F1F2given

By [F1] the group $G$ is reductive. Since $G$ is semisimple, [F2] shows that $Z (G)$ is finite, so its largest central torus $Z (G)_{t}$ is trivial, and the decomposition $G = Z (G)_{t} \cdot G^{'}$ of [F2] gives $G = G^{'} = [G, G]$ .

2.1F3step 1.1

Let $χ \in X (G)$ . By [F3] the character $χ$ is trivial on every commutator, hence on $[G, G]$ , which is all of $G$ by step 1.1; therefore $χ$ is the trivial character.

3.1F4step 2.1

By [F4] a one-dimensional rational representation of $G$ is given by a character on $G$ ; by step 2.1 every such representation is trivial.

4.1step 1.1step 3.1∎

Steps 1.1 and 3.1 give both $G = [G, G]$ and $X (G) = 0$ , and the displayed equivalence with triviality of all one-dimensional rational representations.

Remarks

No splitness is used; the argument applies to every semisimple algebraic group over $k$ .
The commutator identity used for characters is the only place where the commutativity of $G_{m}$ enters; it makes every character factor through the abelianization $G / [G, G]$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

The Casimir element of a rational representation is an endomorphism of G-modules

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $0$ and let $(V, r)$ be a finite-dimensional rational representation with $ρ : g \to g l (V)$ the derived representation and $\overset{ˉ}{g} = ρ (g)$ . If $\overset{ˉ}{g} \neq 0$ , then: (a) $\overset{ˉ}{g}$ is a semisimple Lie algebra and the trace form $B_{ρ}$ of the faithful representation of $\overset{ˉ}{g}$ on $V$ is nondegenerate; (b) the Casimir element $Ω_{B_{ρ}} \in U (\overset{ˉ}{g})$ of $B_{ρ}$ defines a $G$ -module endomorphism $c_{V} : V \to V$ (Casimir operator relative to an invariant form, The Casimir operator is basis-independent and intertwining); (c) $tr (c_{V} ∣_{V}) = \dim_{k} \overset{ˉ}{g}$ . Here $U (\overset{ˉ}{g})$ is the universal enveloping algebra, and $c_{V}$ is the action of $Ω_{B_{ρ}}$ on $V$ induced by the inclusion $\overset{ˉ}{g} ↪ g l (V)$ .

Facts & Assumptions

Given: A semisimple algebraic group $G$ over a characteristic-zero field $k$ with $g = Lie (G)$ , a finite-dimensional rational representation $(V, r)$ with differential $ρ : g \to g l (V)$ , and $\overset{ˉ}{g} = ρ (g) \neq 0$ .

[F1]

Quotients of semisimple Lie algebras. $g$ is semisimple (The Lie algebra of a semisimple group in characteristic zero is semisimple), and every quotient of a finite-dimensional semisimple characteristic-zero Lie algebra is semisimple (Ideals and quotients of semisimple Lie algebras); hence $\overset{ˉ}{g} ≅ g / \ker ρ$ is semisimple.

[F2]

Nondegenerate trace form. The representation $\overset{ˉ}{g} ↪ g l (V)$ is faithful and finite-dimensional, so its trace form $B_{ρ} (x, y) = tr (x y)$ for $x, y \in \overset{ˉ}{g}$ is nondegenerate and invariant (Trace forms of faithful representations of semisimple Lie algebras are nondegenerate).

[F3]

Casimir element. For a semisimple Lie algebra with nondegenerate invariant form $B$ , a basis $(e_{i})$ and the $B$ -dual basis $(e_{i}^{'})$ , the element $Ω_{B_{ρ}} = \sum_{i} e_{i} e_{i}^{'} \in U (\overset{ˉ}{g})$ is independent of the basis. Its action $c_{V} = \sum_{i} e_{i} \circ e_{i}^{'} \in {End}_{k} (V)$ is an endomorphism of $V$ as a $\overset{ˉ}{g}$ -module, and $tr (c_{V} ∣_{V}) = \sum_{i} B (e_{i}, e_{i}^{'}) = \dim \overset{ˉ}{g}$ (Casimir operator relative to an invariant form, The Casimir operator is basis-independent and intertwining).

[F4]

Endomorphisms of $V$ . The space ${End}_{k} (V) ≅ V^{*} \otimes_{k} V$ is a finite-dimensional rational representation of $G$ with $(g \cdot f) (v) = r (g) f (r (g)^{- 1} v)$ , whose fixed points are exactly the $G$ -module endomorphisms of $V$ (Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational, Rational representations and comodules of an affine group scheme).

[F5]

Lie-stable subspaces are stable. Since $k$ has characteristic $0$ and $G$ is connected and smooth, a subspace $W$ of a rational representation with $g W \subseteq W$ is $G$ -stable (Lie algebras of subspace stabilizers and Lie-stable subspaces).

[F6]

Semisimple groups have no characters. $X (G) = 0$ , so a one-dimensional rational representation of the semisimple group $G$ is trivial (Semisimple groups are perfect and have no nontrivial characters).

Proof

technique · direct

1.1F1F2given

The image $\overset{ˉ}{g} = ρ (g)$ is a quotient of $g$ by the ideal $\ker ρ$ , so it is semisimple by [F1], and $B_{ρ}$ is the trace form of the faithful finite-dimensional representation of $\overset{ˉ}{g}$ on $V$ , hence nondegenerate by [F2]. This is (a).

2.1F3step 1.1

By [F3], $Ω_{B_{ρ}} = \sum_{i} e_{i} e_{i}^{'} \in U (\overset{ˉ}{g})$ is basis-independent. Its action on $V$ is $c_{V} = \sum_{i} e_{i} \circ e_{i}^{'} \in {End}_{k} (V)$ , where the $e_{i}, e_{i}^{'}$ are already operators in $\overset{ˉ}{g} \subseteq g l (V)$ . This operator commutes with every $x \in \overset{ˉ}{g}$ and has trace $\sum_{i} B_{ρ} (e_{i}, e_{i}^{'}) = \dim \overset{ˉ}{g}$ .

3.1F4F5F6step 2.1

The line $W = k c_{V}$ inside the rational representation ${End}_{k} (V) ≅ V^{*} \otimes_{k} V$ of [F4] is annihilated by $g$ , because the infinitesimal action is $x \cdot f = [ρ (x), f]$ and step 2.1 gives $[ρ (x), c_{V}] = 0$ ; in particular $g W \subseteq W$ . By [F5] the line $W$ is $G$ -stable, and by [F6] the action of $G$ on the one-dimensional representation $W$ is trivial. Hence $c_{V}$ is a fixed point of the action on ${End}_{k} (V)$ , so by [F4] it is a $G$ -module endomorphism of $V$ . This is (b).

3.2step 2.1

The trace identity $tr (c_{V} ∣_{V}) = \dim_{k} \overset{ˉ}{g}$ of step 2.1 is (c).

4.1step 1.1step 3.1step 3.2∎

Steps 1.1, 3.1 and 3.2 establish (a), (b) and (c).

Remarks

The point of the lemma is that the Casimir operator, which a priori is only an endomorphism of $g$ -modules, is fixed by the whole connected group in characteristic zero: the line it spans is a one-dimensional rational representation of the semisimple group $G$ , and $X (G) = 0$ .
If $\overset{ˉ}{g} = 0$ , the representation is trivial by the characteristic-zero connected equal-Lie subgroup criterion. The empty-sum convention defines both the Casimir element and its operator as zero; the hypothesis $\overset{ˉ}{g} \neq 0$ ensures a nonzero trace and a nonzero line $k c_{V}$ in step 3.1.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Semisimple groups in characteristic zero are linearly reductive

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $0$ . Then every finite-dimensional rational representation of $G$ is semisimple; equivalently, $G$ is linearly reductive (Rational representations and comodules of an affine group scheme, Simple and semisimple rational representations).

Facts & Assumptions

Given: A semisimple algebraic group $G$ over a characteristic-zero field $k$ and a finite-dimensional rational representation $(V, r)$ .

[F1]

Descent of semisimplicity. If $(V_{k^{'}}, r_{k^{'}})$ is semisimple for a field extension $k^{'} \supseteq k$ , then $(V, r)$ is semisimple (Semisimplicity of rational representations descends along field extensions).

[F2]

Reduction to codimension one. If $X (G) = 0$ , then the following are equivalent: (a) every finite-dimensional rational representation of $G$ is semisimple; (b) every subrepresentation of codimension one is a direct summand; (c) every simple subrepresentation of codimension one is a direct summand (Complete reducibility reduces to splitting codimension-one simple submodules).

[F3]

No characters. $X (G) = 0$ , and the same holds after any field extension (Semisimple groups are perfect and have no nontrivial characters).

[F4]

Casimir endomorphism. For a finite-dimensional rational representation $(V, r)$ with $\overset{ˉ}{g} = ρ (g) \neq 0$ , the Casimir element $c_{V}$ of the nondegenerate trace form $B_{ρ}$ is a $G$ -module endomorphism of $V$ with $tr (c_{V} ∣_{V}) = \dim_{k} \overset{ˉ}{g}$ (The Casimir element of a rational representation is an endomorphism of G-modules).

[F5]

Trivial derived action. If $ρ (g) = 0$ for a rational representation of the connected group $G$ , then $V$ is the trivial representation: $\ker r$ is a closed subgroup scheme with $Lie (\ker r) = \ker ρ = g$ , it is smooth in characteristic $0$ , and a connected group equals its smooth closed subgroup with the same Lie algebra (The Lie functor: exactness, fixed points and generation, Rational representations and comodules of an affine group scheme, Cartier's theorem: affine group schemes in characteristic zero are smooth).

[F6]

An eigenvalue exists. A linear operator on a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue. (Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue)

[F7]

Every finite subset of a rational representation lies in a finite-dimensional subrepresentation. (Every element of a comodule lies in a finite-dimensional subcomodule)

[A1]

Under AC, every nonempty poset whose chains have upper bounds has a maximal element. (Zorn's lemma)

Proof

technique · direct

1.1F1given

It suffices to prove the assertion after extending scalars to an algebraic closure $k^{a}$ of $k$ : if every finite-dimensional representation of $G_{k^{a}}$ is semisimple, then [F1] gives semisimplicity of every finite-dimensional representation of $G$ . We therefore assume in the rest of the proof that $k$ is algebraically closed.

2.1F2F3F5step 1.1algebra

For the algebraically closed field $k$ one has $X (G) = 0$ by [F3], so by [F2] it suffices to verify condition (c): every simple subrepresentation $W$ of codimension one in a finite-dimensional $V$ is a direct summand. If $V$ is trivial, any finite-dimensional linear complement to $W$ is a $G$ -subrepresentation, so the condition holds in every dimension, including zero. Otherwise $\overset{ˉ}{g} = ρ (g) \neq 0$ by [F5], and we may use its Casimir operator.

3.1F3F4step 2.1

Let $W$ be a simple subrepresentation of codimension one in a nonzero finite-dimensional $V$ with $\overset{ˉ}{g} \neq 0$ , and let $c_{V}$ be the Casimir endomorphism of [F4]. The quotient $V / W$ is a one-dimensional rational representation, hence trivial by [F3]; therefore $g V \subseteq W$ . Since $c_{V}$ is a sum of products $ρ (x) ρ (y)$ with $x, y \in g$ , it follows that $c_{V} (V) \subseteq g V \subseteq W$ .

4.1F4F6step 3.1algebra

The restriction $c_{V} ∣_{W}$ is a $G$ -module endomorphism by [F4]. Since $k$ is algebraically closed and $W$ is nonzero finite-dimensional, it has an eigenvalue $a \in k$ . The kernel of $c_{V} ∣_{W} - a {id}_{W}$ is nonzero and is a $G$ -subrepresentation, because that difference is $G$ -equivariant. Simplicity of $W$ makes this kernel all of $W$ . Hence $c_{V} ∣_{W} = a {id}_{W}$ , by the eigenvalue-kernel argument of Schur's lemma applied directly to the rational $G$ -module.

5.1F4step 3.1step 4.1

The scalar $a$ is nonzero: $c_{V}$ maps $V$ into $W$ , so $tr (c_{V} ∣_{V}) = tr (c_{V} ∣_{W}) = a \dim W$ , while [F4] gives $tr (c_{V} ∣_{V}) = \dim_{k} \overset{ˉ}{g} \neq 0$ ; hence $a \neq 0$ . Therefore $c_{V} ∣_{W}$ is invertible, $\ker c_{V}$ intersects $W$ trivially, $c_{V} (V) = W$ , and $\dim \ker c_{V} = \dim V - \dim W = 1$ . Since $c_{V}$ is a $G$ -module endomorphism by [F4], its kernel is a $G$ -submodule, and $V = W \oplus \ker c_{V}$ exhibits $W$ as a direct summand.

6.1F2step 1.1step 5.1

Condition (c) of [F2] holds for every finite-dimensional $V$ by step 5.1, so by [F2] every finite-dimensional rational representation is semisimple; by step 1.1 this descends to the original field.

7.1F7A1step 6.1algebra∎

For an arbitrary rational representation $M$ , order by inclusion the sets of simple submodules whose sum is direct. The empty set is such a family, and the union of a chain is such a family because each finite relation occurs in one member of the chain. By [A1] choose a maximal family with sum $S$ . If $S \neq M$ , [F7] gives a finite-dimensional submodule $W$ containing a vector outside $S$ . By step 6.1, $W$ is a finite direct sum of simple submodules. Since $W$ is not contained in $S$ , one such summand $C$ is not contained in $S$ ; simplicity gives $C \cap S = 0$ , so adjoining $C$ extends the family, a contradiction. Therefore $M = S$ is a direct sum of simple submodules, establishing linear reductivity.

Remarks

The proof is Milne's Proposition 22.41; the independence from the base field, the reduction to codimension-one simple submodules and the Casimir argument are the three inputs (Milne 22.39, 22.40 and the characteristic-zero section).
The hypothesis that $G$ is semisimple enters twice: through $X (G) = 0$ and through the nonvanishing of the Casimir trace $\dim \overset{ˉ}{g}$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Complete reducibility of rational modules in characteristic zero

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be a connected reductive algebraic group over a field $k$ of characteristic $0$ (in particular, let $G$ be any split reductive group over such a field). Then the following are equivalent: (a) $G$ is reductive; (b) every finite-dimensional rational representation of $G$ is semisimple; (c) some faithful finite-dimensional rational representation of $G$ is semisimple (Rational representations and comodules of an affine group scheme, Radical, unipotent radical, semisimple and reductive algebraic groups, Simple and semisimple rational representations). In particular $G$ is linearly reductive, so every rational representation is a direct sum of simple representations.

Facts & Assumptions

Given: A connected reductive algebraic group $G$ over a characteristic-zero field $k$ , its unipotent radical $U = R_{u} (G)$ (Radical, unipotent radical, semisimple and reductive algebraic groups) and its derived subgroup $G^{'} = [G, G]$ (The derived subgroup, the derived series and solvable algebraic groups).

[F1]

A reductive group is the almost product $G = Z (G)_{t} G^{'}$ of its largest central torus and its semisimple derived group. In characteristic $0$ , Cartier makes the centre smooth, so $Z (G)^{\circ} = Z (G)_{t}$ . (Centre, radical and semisimple quotient of a reductive group, Groups of multiplicative type and tori, Cartier's theorem: affine group schemes in characteristic zero are smooth)

[F3]

Semisimple groups in characteristic zero. Every finite-dimensional rational representation of a semisimple group over a characteristic-zero field is a direct sum of simple subrepresentations (Semisimple groups in characteristic zero are linearly reductive).

[F4]

Faithful representations exist. Every affine group scheme of finite type over a field has a faithful finite-dimensional rational representation (A finitely generated affine group scheme has a faithful finite-dimensional representation).

[F5]

Unipotent fixed vectors. A unipotent affine algebraic group has a nonzero fixed vector in every nonzero rational representation; equivalently, every simple representation of a unipotent group is one-dimensional with trivial action (Unipotent algebraic groups and unipotent representations).

[F7]

Descent of semisimplicity. If a finite-dimensional rational module becomes semisimple after a field extension, it is semisimple over the original field (Semisimplicity of rational representations descends along field extensions).

[F8]

Weights of a split torus. A rational representation of a split torus decomposes as the direct sum of its character weight spaces (Representations of diagonalizable groups split into character eigenspaces).

[F9]

Every finite subset of a rational representation is contained in a finite-dimensional subrepresentation. (Every element of a comodule lies in a finite-dimensional subcomodule)

[A1]

Under AC, every nonempty poset whose chains have upper bounds has a maximal element. (Zorn's lemma)

Proof

technique · direct

1.1F4given

(b) $\Rightarrow$ (c). By [F4] there is a faithful finite-dimensional rational representation of $G$ ; if (b) holds, that representation is semisimple, so (c) holds.

1.2F5given

(c) $\Rightarrow$ (a). Let $V$ be a faithful semisimple finite-dimensional representation and put $U = R_{u} (G)$ , which is unipotent and normal in $G$ . For every simple subrepresentation $S \subseteq V$ the fixed space $S^{U}$ is nonzero by [F5]; it is a $G$ -subrepresentation because $U$ is normal, so simplicity gives $S^{U} = S$ , that is, $U$ acts trivially on $S$ . Hence $U$ acts trivially on $V$ , and faithfulness forces $U = 1$ . Over the perfect field $k$ of characteristic $0$ , triviality of the unipotent radical is exactly reductivity of $G$ , so (a) holds.

1.3F1F8given

(a) $\Rightarrow$ (b), reduce to a split central torus. Extend scalars to an algebraic closure $\overset{ˉ}{k}$ ; it is enough first to prove that $V_{\overset{ˉ}{k}}$ is semisimple. Apply [F1] to $G_{\overset{ˉ}{k}}$ : its largest central torus $Z (G_{\overset{ˉ}{k}})_{t}$ is a split torus, and its derived subgroup $G_{\overset{ˉ}{k}}^{'}$ is semisimple. By [F8], $V_{\overset{ˉ}{k}}$ is the direct sum of the character weight spaces for $Z (G_{\overset{ˉ}{k}})_{t}$ . Since this torus is central in $G_{\overset{ˉ}{k}}$ , each weight space is stable under $G_{\overset{ˉ}{k}}^{'}$ .

2.1F3F7step 1.3

(a) $\Rightarrow$ (b), decompose the weight spaces. Each weight space from step 1.3 is a finite-dimensional rational representation of $G_{\overset{ˉ}{k}}^{'}$ , so [F3] decomposes it into simple $G_{\overset{ˉ}{k}}^{'}$ -submodules. The central torus acts on each whole weight space by its character, so every such simple submodule is stable under both $Z (G_{\overset{ˉ}{k}})_{t}$ and $G_{\overset{ˉ}{k}}^{'}$ , hence under their product $G_{\overset{ˉ}{k}}$ . Thus $V_{\overset{ˉ}{k}}$ is semisimple. By [F7] semisimplicity descends from $\overset{ˉ}{k}$ to $k$ , proving (a) $\Rightarrow$ (b) over the original field.

3.1F9A1step 1.1step 1.2step 2.1∎

Steps 1.1, 1.2 and 2.1 prove the equivalence of (a), (b) and (c). If (b) holds, an arbitrary rational representation $V$ is the union of its finite-dimensional subrepresentations by [F9], each of which is a direct sum of simple subrepresentations; the sets of simple subrepresentations whose sum is direct form a nonempty poset under inclusion. The union of a chain is again such a family, since every finite relation occurs in one chain member. By [A1] choose a maximal family with sum $S \subseteq V$ . If $S \neq V$ , [F9] gives a finite-dimensional subrepresentation $W$ containing a vector outside $S$ . A simple summand $C$ of $W$ is then not contained in $S$ , and simplicity gives $C \cap S = 0$ , so adjoining $C$ extends the family, a contradiction. Hence $V = S$ is a direct sum of simple representations, that is, $G$ is linearly reductive.

Remarks

The three implications are Milne's proof of Theorem 22.42: the structure $G = Z (G)_{t} \cdot G^{'}$ reduces the reductive case to the multiplicative-type and semisimple cases, while the converse uses that a unipotent radical acts trivially on every simple module.
The final Zorn argument extends the finite-dimensional statement to arbitrary rational representations; the Axiom of Choice is declared and used there in addition to the inherited AC premises of the structural suppliers.

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Weights, dominant weights and the highest-weight order of a rational representation

Definition

Let $k$ be a field, let $(G, T)$ be a split reductive group over $k$ with Weyl group $W$ and root datum $(X, Φ, α \mapsto α^{\lor})$ , $X = X (T)$ , and let $B \supseteq T$ be a Borel subgroup with system of positive roots $Φ^{+}$ and base $Δ$ (Split reductive groups, Roots and root groups of a split reductive group, Root subgroups of a split reductive group, Character and cocharacter lattices of a split torus). For a rational representation $(V, r)$ of $G$ (Rational representations and comodules of an affine group scheme) the $T$ -weight decomposition is $V = ⨁_{χ \in X} V_{χ}$ with $V_{χ} = {v \in V : ρ (v) = v \otimes e_{χ}}$ (Representations of diagonalizable groups split into character eigenspaces); the weights of $(V, r)$ are the characters $χ$ with $V_{χ} \neq 0$ , and $V_{χ}$ is the weight space of $χ$ . An element $λ \in X$ is dominant if $⟨ λ, α^{\lor} ⟩ \geq 0$ for every $α \in Φ^{+}$ (equivalently, for every $α \in Δ$ ); write $X^{+}$ for the set of dominant weights. Define a partial order on $X$ by $λ \geq μ$ if and only if $λ - μ = \sum_{α \in Δ} m_{α} α$ with $m_{α} \in Z_{\geq 0}$ . The fundamental weights $ω_{i}$ ( $i \in Δ$ ) are the unique elements of $Q Φ \subseteq X \otimes_{Z} Q$ dual to the simple coroots, $⟨ ω_{i}, α_{j}^{\lor} ⟩ = δ_{i j}$ ; they form a basis of $Q Φ$ and every $λ \in X \otimes Q$ decomposes as $λ = \sum_{i} ⟨ λ, α_{i}^{\lor} ⟩ ω_{i} + λ_{0}$ with $λ_{0}$ pairing to $0$ with every coroot (Combinatorics of a reduced root datum).

Remarks

Conventions. Dominance is tested on the positive system $Φ^{+}$ , which is the system of roots of $B$ ; the equivalence of testing on $Φ^{+}$ or on the base $Δ$ holds for coroot pairings as follows. Average an inner product on $X \otimes R$ over the finite Weyl group of Combinatorics of a reduced root datum. Each root reflection is then orthogonal, so this inner product identifies $α^{\lor}$ with $2 α / (α, α)$ . If $α = \sum_{i} n_{i} α_{i}$ is positive, then $α^{\lor} = \sum_{i} n_{i} (α_{i}, α_{i}) / (α, α) α_{i}^{\lor}$ , a nonnegative combination of simple coroots. Thus testing the simple coroots suffices. The order is generated by the positive simple roots, so $λ \geq μ$ is the relation used for highest weights.
Existence and uniqueness. In that inner product the simple-coroot functionals are independent on the root span, since the simple roots form a basis. Their pairing matrix is rational and invertible, so their dual vectors $ω_{i}$ lie uniquely in $Q Φ$ . Subtracting $\sum_{i} ⟨ λ, α_{i}^{\lor} ⟩ ω_{i}$ from $λ$ gives the displayed $λ_{0}$ , which annihilates all coroots by the preceding coroot expansion. Specifying the root span is essential when the central torus is nontrivial.
Fundamental weights need not lie in $X$ . The $ω_{i}$ are elements of the rational vector space $X \otimes Q$ ; the correction term $λ_{0}$ in the displayed decomposition lies in the annihilator of the coroot lattice, and it need not belong to $X$ , nor need $\sum_{i} ⟨ λ, α_{i}^{\lor} ⟩ ω_{i}$ lie in $X$ for an arbitrary reductive root datum. This is why the classification below is proved through the product-with-a-torus and central-isogeny route rather than by adding a lattice correction term directly.
No Choice. The definition only records eigenspace decompositions and lattice dualities of the cited suppliers; no choice principle is used.

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Primitive vectors for a Borel pair

Definition

Let $(G, T)$ be a split reductive group, $B \supseteq T$ a Borel subgroup with unipotent radical $U = B_{u}$ and roots $Φ$ (Borel subgroups, maximal tori and Borel pairs, Unipotent algebraic groups and unipotent representations, Roots and root groups of a split reductive group, Root subgroups of a split reductive group). Let $(V, r)$ be a rational representation (Rational representations and comodules of an affine group scheme). A nonzero vector $v \in V$ is primitive (for the pair $(B, T)$ ) if it spans a $B$ -stable line. Such a vector is fixed by $U$ and is a $T$ -eigenvector: the action on its one-dimensional line restricts to a character of $T$ , and unipotence forces the action of $U$ on that line to be trivial. Assuming the Axiom of Choice (The Axiom of Choice) for the cited split-Borel structure, multiplication is an isomorphism $U ⋊ T \to B$ , as justified in the Remarks below. Consequently the converse holds as well: a nonzero $U$ -fixed $T$ -eigenvector is primitive. The weight of a primitive vector is the character $λ \in X (T)$ with $t \cdot v = λ (t) v$ for all $t$ and all $k$ -algebras; for a finite-dimensional $V$ it is the weight $λ$ with $v \in V_{λ}$ (Weights, dominant weights and the highest-weight order of a rational representation).

Remarks

Weight and converse. Restriction to a fixed line gives a unique character of $T$ , so its weight is well defined without a choice principle. The forward implication uses only the defining fixed-vector property of a unipotent group applied to that line. For the converse, under AC, the root-group coordinates give $\dim U = ∣ Φ^{+} ∣$ , while the adjoint weight decomposition gives $\dim B = \dim T + ∣ Φ^{+} ∣$ . The intersection $U \cap T$ is trivial by A subgroup that is both unipotent and diagonalizable is trivial. Since $U$ is normal in $B$ , multiplication gives a homomorphism $U ⋊ T \to B$ with trivial scheme kernel; the exact-image theorem Group images are exact kernel quotients and preserve affine smooth connected properties identifies its source with a smooth connected closed image of the same dimension as the smooth connected group $B$ . Thus that image is $B$ , proving $B = U ⋊ T$ . A $U$ -fixed $T$ -eigenline is consequently $B$ -stable.
Normalisation. The Borel subgroup is the one used to define the positive system $Φ^{+}$ , so a primitive vector is fixed by the unipotent group of positive root subgroups. The opposite unipotent group $U^{-}$ generates the big cell with $B$ and is used only in the proofs of the weight statements.
Choice scope. The definition by a $B$ -stable line, its weight, and the forward implication above are choice-free. AC is inherited only for the supplemental structural converse and the root-group facts used to describe the opposite big cell.

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Expansion of a root-group translate of a weight vector

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(V, r)$ be a rational representation of a split reductive group $(G, T)$ , let $v \in V_{λ}$ be a weight vector, and let $α \in Φ$ with root-group isomorphism $u_{α} : G_{a} \to U_{α}$ (Root subgroups of a split reductive group). These coordinates satisfy $t u_{α} (c) t^{- 1} = u_{α} (α (t) c)$ over every $k$ -algebra. Then there are vectors $v_{i} \in V_{λ + i α}$ ( $i \geq 1$ ), only finitely many nonzero, such that $u_{α} (c) \cdot v = v + \sum_{i \geq 1} c^{i} v_{i}$ for every $c$ (and every $k$ -algebra after base change). In particular the orbit map $c \mapsto u_{α} (c) \cdot v$ is polynomial with constant term $v$ and higher coefficients in the stated weight spaces.

Facts & Assumptions

Given: A split reductive group $(G, T)$ with root $α \in Φ$ , the root-group isomorphism $u_{α} : G_{a} \to U_{α}$ , a rational representation $(V, r)$ and a weight vector $v \in V_{λ}$ (Weights, dominant weights and the highest-weight order of a rational representation).

[F1]

Finite-dimensional orbit. The vector $v$ lies in a finite-dimensional subcomodule $W \subseteq V$ ; the action of $U_{α}$ preserves $W$ , so $u_{α} (c) \cdot v \in W$ for all $c$ (Every element of a comodule lies in a finite-dimensional subcomodule, Rational representations and comodules of an affine group scheme).

[F2]

Rational action is polynomial. For a finite-dimensional rational representation $W$ of $G_{a}$ the matrix coefficients of $c \mapsto r (u_{α} (c))$ are polynomial functions of $c$ ; hence $c \mapsto u_{α} (c) \cdot v$ is given by a polynomial in $c$ with values in $W$ , and a basis of $W$ exhibits it as $u_{α} (c) \cdot v = \sum_{i \geq 0} c^{i} v_{i}$ with $v_{i} \in W$ , only finitely many nonzero (Rational representations and comodules of an affine group scheme).

[F3]

Conjugation formula. For $t \in T (R)$ and $c \in R$ one has $t u_{α} (c) t^{- 1} = u_{α} (α (t) c)$ (Root subgroups of a split reductive group, Roots and root groups of a split reductive group). To justify the formula from the supplied $T$ -stability and Lie weight, work with the universal torus point over $A = k [x_{1}^{\pm 1}, \dots, x_{r}^{\pm 1}]$ , a domain. Conjugation induces a polynomial automorphism of $A [c]$ fixing $c = 0$ ; its polynomial inverse and the degree-of-composition identity over a domain force it to be $c \mapsto a (t) c$ , with $a (t)$ a unit. Its differential at $c = 0$ is the adjoint character $α (t)$ , hence $a (t) = α (t)$ . This universal identity specializes to every $R$ , including nonreduced base algebras.

[F4]

Weight vectors. $t \cdot v = λ (t) v$ for all $t \in T (R)$ , and the weight spaces $V_{χ}$ are the eigenspaces of the $T$ -action (Weights, dominant weights and the highest-weight order of a rational representation).

Proof

technique · direct

1.1F1F2given

By [F1] and [F2] there is a finite expansion $u_{α} (c) \cdot v = \sum_{i \geq 0} c^{i} v_{i}$ with $v_{i} \in W$ and only finitely many nonzero. Setting $c = 0$ gives $v_{0} = v$ .

2.1F3F4step 1.1

For $t \in T (R)$ one has $t \cdot (u_{α} (c) \cdot v) = (t u_{α} (c) t^{- 1}) \cdot (t \cdot v) = u_{α} (α (t) c) \cdot (λ (t) v) = λ (t) \sum_{i \geq 0} α (t)^{i} c^{i} v_{i}$ using [F3] and [F4].

3.1F4step 1.1step 2.1

On the other hand $t \cdot (u_{α} (c) \cdot v) = \sum_{i \geq 0} c^{i} (t \cdot v_{i})$ by linearity of the action over $R [c]$ . Comparing coefficients of $c^{i}$ in the two polynomial expressions for all $R$ -points $t$ shows $t \cdot v_{i} = λ (t) α (t)^{i} v_{i}$ for every $i$ , that is, $v_{i} \in V_{λ + i α}$ : the character $χ_{i}$ with $t \cdot v_{i} = χ_{i} (t) v_{i}$ satisfies $χ_{i} (t) = λ (t) α (t)^{i}$ for all $t$ and all $R$ . Together with step 1.1 this gives the asserted expansion.

4.1step 1.1step 3.1∎

Since $v_{0} = v$ and $v_{i} \in V_{λ + i α}$ for $i \geq 1$ with only finitely many nonzero, the orbit map $c \mapsto u_{α} (c) \cdot v$ is polynomial with constant term $v$ and higher coefficients in the stated weight spaces.

Remarks

The base-change clause of the statement is included because the computation is carried out for an arbitrary $k$ -algebra $R$ of values of $c$ and points $t \in T (R)$ ; the expansion is the same polynomial for every $R$ .
If $v_{i} = 0$ for all $i \geq 1$ the vector $v$ is fixed by $U_{α}$ ; the vanishing of all higher coefficients is what makes a primitive vector fixed by every positive root group.

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Tensor products of primitive vectors

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(V, r)$ and $(V^{'}, r^{'})$ be rational representations of a split reductive group $(G, T)$ with primitive vectors $v, v^{'}$ of weights $λ, λ^{'}$ (Primitive vectors for a Borel pair). Then $v \otimes v^{'}$ is a primitive vector of $V \otimes V^{'}$ of weight $λ + λ^{'}$ . Consequently tensor powers and tensor products of primitive vectors are primitive, with the summed weights.

Facts & Assumptions

Given: Rational representations $(V, r)$ , $(V^{'}, r^{'})$ of the split reductive group $(G, T)$ with primitive vectors $v \in V$ , $v^{'} \in V^{'}$ and weights $λ, λ^{'}$ , and the unipotent radical $U = B_{u}$ of a Borel subgroup $B \supseteq T$ .

[F1]

Tensor coactions. The representation/comodule dictionary applies to arbitrary vector spaces. For finite-dimensional pairs, Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational gives the tensor coaction explicitly. The same formula from the two finite coaction expansions on each elementary tensor is defined in arbitrary dimension by the tensor universal property; its counit and coassociativity identities follow from those of the two factors and multiplicativity of the Hopf-algebra counit and coproduct. This is verified directly in step 1.1. (Rational representations and comodules of an affine group scheme, Universal property of the tensor product for balanced maps into abelian groups)

[F2]

Primitive vectors. A nonzero vector is primitive of weight $χ$ exactly when it is fixed by $U$ and is a $T$ -eigenvector with character $χ$ ; in particular $u \cdot v = v$ and $t \cdot v = λ (t) v$ for all $R$ -points $u \in U (R)$ , $t \in T (R)$ , and likewise for $v^{'}$ (Primitive vectors for a Borel pair, Rational representations and comodules of an affine group scheme).

[F3]

Nonzero elementary tensors. Under the existing AC premise every vector space has a basis (Every vector space has a basis). For a nonzero vector choose a nonzero coordinate in that basis and rescale its coordinate functional to value $1$ . Thus there are $ϕ \in V^{*}$ and $ϕ^{'} \in (V^{'})^{*}$ with $ϕ (v) = ϕ^{'} (v^{'}) = 1$ . Their bilinear product defines a linear functional on $V \otimes V^{'}$ taking $v \otimes v^{'}$ to $1$ , by the tensor universal property. (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ , Universal property of the tensor product for balanced maps into abelian groups)

Proof

technique · direct

1.1F1F3givenalgebra

For finite-dimensional $V, V^{'}$ , [F1] supplies the tensor representation. In arbitrary dimension, write $ρ (v) = \sum_{i} v_{i} \otimes a_{i}$ and $ρ^{'} (v^{'}) = \sum_{j} v_{j}^{'} \otimes b_{j}$ . Each expansion is finite, even for arbitrary-dimensional representations. The formula $c (v \otimes v^{'}) = \sum_{i, j} v_{i} \otimes v_{j}^{'} \otimes a_{i} b_{j}$ is induced by a bilinear map, hence defines a linear coaction candidate by [F1]. Its counit sends this expression to $v \otimes v^{'}$ because $ε (a_{i} b_{j}) = ε (a_{i}) ε (b_{j})$ . Its two iterated coactions agree because the coactions of both factors are coassociative and $Δ (a_{i} b_{j}) = Δ (a_{i}) Δ (b_{j})$ . Thus the representation/comodule dictionary gives a rational representation on $V \otimes V^{'}$ ; evaluating at any $R$ -point $g$ gives $g (v \otimes v^{'}) = g v \otimes g v^{'}$ . The two coordinate functionals of [F3] evaluate $v \otimes v^{'}$ to $1$ , so this vector is nonzero. This proves all tensor inputs needed below in arbitrary dimension.

1.2F1F2

For every $R$ -point $t \in T (R)$ one has $t \cdot (v \otimes v^{'}) = t \cdot v \otimes t \cdot v^{'} = λ (t) λ^{'} (t) (v \otimes v^{'}) = (λ + λ^{'}) (t) (v \otimes v^{'})$ , so $v \otimes v^{'}$ is a $T$ -eigenvector of weight $λ + λ^{'}$ .

2.1F1F2step 1.1

For every $R$ -point $u \in U (R)$ one has $u \cdot (v \otimes v^{'}) = u \cdot v \otimes u \cdot v^{'} = v \otimes v^{'}$ , so $v \otimes v^{'}$ is fixed by $U$ .

3.1F2step 1.1step 2.1step 1.2

By [F2] a nonzero $U$ -fixed $T$ -eigenvector of weight $λ + λ^{'}$ is primitive of that weight, so $v \otimes v^{'}$ is primitive of weight $λ + λ^{'}$ .

4.1step 3.1algebra∎

Iterating the construction gives that tensor powers $v^{\otimes m}$ are primitive of weight $m λ$ and tensor products of finitely many primitive vectors are primitive with the sum of the weights. For $m = 0$ , the empty tensor is $1$ in the trivial module $k$ , primitive of weight $0$ .

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The normalizer of the torus permutes weight spaces

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(V, r)$ be a rational representation of a split reductive group $(G, T)$ and let $n \in N_{G} (T) (R)$ for a $k$ -algebra $R$ . If $v \in V_{λ}$ , then $n \cdot (v \otimes 1) \in (V \otimes_{k} R)_{n λ}$ , where $n λ : T_{R} \to G_{m, R}$ is the character $t \mapsto λ (n^{- 1} t n)$ , tested over every $R$ -algebra. If $R$ is disconnected, this character can vary between components; the target denotes its eigensubmodule over $R$ . Consequently, for every $w \in W (G, T)$ the weight spaces $V_{λ}$ and $V_{w λ}$ have the same dimension, and the set of weights of $(V, r)$ is stable under $W$ (The Weyl group, Borel subgroups and chambers, The root datum of a split reductive group).

Facts & Assumptions

Given: A rational representation $(V, r)$ of the split reductive group $(G, T)$ , a weight vector $v \in V_{λ}$ and a point $n \in N_{G} (T) (R)$ for a $k$ -algebra $R$ .

[F1]

The character $n λ$ . For $n \in N_{G} (T) (R)$ the assignment $t \mapsto λ (n^{- 1} t n)$ is a character $n λ \in X (T)_{R}$ of $T_{R}$ : it is a morphism in $t$ and multiplicative because conjugation is a group homomorphism (The root datum of a split reductive group, Weights, dominant weights and the highest-weight order of a rational representation).

[F2]

Weight vectors. For $t \in T (R)$ and $v \in V_{λ}$ one has $t \cdot v = λ (t) v$ , and the weight spaces are the eigenspaces of the $T$ -action (Weights, dominant weights and the highest-weight order of a rational representation).

[F3]

Weyl group. $W (G, T) = N_{G} (T) (k) / T (k)$ , and every $w \in W$ has representatives $n \in N_{G} (T) (k)$ and $n^{- 1}$ for $w^{- 1}$ (The Weyl group, Borel subgroups and chambers, The root datum of a split reductive group).

Proof

technique · direct

1.1F1F2given

For $t \in T (R)$ one computes $t \cdot (n \cdot v) = n \cdot ((n^{- 1} t n) \cdot v) = n \cdot (λ (n^{- 1} t n) v) = λ (n^{- 1} t n) (n \cdot v) = (n λ) (t) (n \cdot v)$ in $V_{R}$ : the first equality uses that the action is a left action and $n^{- 1} t n \in T (R)$ , and the second uses [F2] and $R$ -linearity of the action of $n$ . Hence $n \cdot (v \otimes 1)$ belongs to the stated eigensubmodule of $V \otimes_{k} R$ , with the calculation valid over every $R$ -algebra.

2.1F3step 1.1

Applying step 1.1 to $n \in N_{G} (T) (R)$ gives a bijection $V_{λ} \otimes_{k} R \to (V \otimes_{k} R)_{n λ}$ , $v \mapsto n \cdot v$ , with inverse given by $n^{- 1}$ ; taking $R = k$ and representatives of $w$ shows $\dim_{k} V_{λ} = \dim_{k} V_{w λ}$ for every $w \in W$ .

3.1step 2.1

Since $V_{χ} \neq 0$ exactly for the weights of $V$ , step 2.1 shows that the set of weights is stable under the action $w ⁣ : λ \mapsto w λ$ of $W$ .

4.1step 1.1step 2.1step 3.1∎

Steps 1.1, 2.1 and 3.1 establish the three assertions.

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The induced coordinate module E(lambda)

Definition

Let $(G, T)$ be a split reductive group over $k$ , and let $B^{0} = B^{-}$ be a provided opposite Borel subgroup. For a provided character of $B^{0}$ whose restriction to $T$ is $λ \in X (T)$ , define $E (λ)$ to be the $k$ -subspace of $O (G)$ (The coordinate Hopf algebra of an affine group scheme) satisfying $f (g b) = f (g) λ (b^{- 1})$ for every $k$ -algebra $R$ , $g \in G (R)$ and $b \in B^{0} (R)$ . Here $λ$ on $B^{0}$ denotes that provided extension. The subspace is stable under the left regular action $(g f) (x) = f (g^{- 1} x)$ and hence is a rational $G$ -module (Rational representations and comodules of an affine group scheme). Its underlying subspace and this action are choice-free. In the source convention this is ${Ind}_{B^{0}}^{G} (k_{λ})$ .

Assuming AC (The Axiom of Choice) for the cited split-Borel structure, $B^{0} = U^{-} ⋊ T$ by the dimension-and-exact-image argument in the Remarks of Primitive vectors for a Borel pair, applied to the opposite Borel. The projection to $T$ extends every $λ \in X (T)$ uniquely to a character of $B^{0}$ trivial on $U^{-}$ . Thus the construction applies to every weight $λ$ of the given split torus. The same structural input gives $B^{-} \cap B = T$ and the open big cells $U^{-} B$ and $U B^{0}$ (Root subgroups of a split reductive group, Bruhat decomposition for a split reductive group, Borel subgroups, maximal tori and Borel pairs).

Remarks

Well-definedness. The defining condition is checked on $R$ -points for every $k$ -algebra $R$ ; since $f$ is a regular function on the affine group scheme $G$ , the condition is an identity of morphisms and the set $E (λ)$ is a $k$ -subspace of $O (G)$ stable under the left regular action: if $f$ satisfies the condition and $g_{0} \in G (R)$ , then $(g_{0} f) (x b) = f (g_{0}^{- 1} x b) = f (g_{0}^{- 1} x) λ (b^{- 1}) = (g_{0} f) (x) λ (b^{- 1})$ for $x \in G (R)$ , $b \in B^{0} (R)$ , so $g_{0} f \in E (λ)$ .
The big cell. The opposite Borel $B^{0} = B^{-}$ is the one appearing in the definition, so an element of $E (λ)$ is a function on $G$ whose restriction to each right $B^{0}$ -coset transforms by $λ^{- 1}$ ; the big cell $U B^{0}$ is the open cell of the Bruhat decomposition associated with $B$ and $B^{0}$ .
Induced module. The identification with ${Ind}_{B^{0}}^{G} (k_{λ}) = {f \in Mor (G, A^{1}) : f (g b) = b^{- 1} f (g)}$ is the source's definition of the induced module; the translation convention $λ (b^{- 1})$ matches the left regular action used here.
Choice scope. For a provided subgroup and character, the equivariance subspace and its left action use no choice principle. AC is inherited only for the supplemental split-Borel projection and big-cell facts above. The comultiplication alone describes right translation, so it is not the coaction label for the left action used here.

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Modules generated by a primitive vector

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(V, r)$ be a rational representation of a split reductive group $(G, T)$ (not necessarily finite-dimensional) that is generated as a $G$ -module by a primitive vector $v$ of weight $λ$ (Primitive vectors for a Borel pair). Then: (a) $V$ is generated as a module over the opposite unipotent group $U^{-}$ by $v$ ; (b) $V = k v \oplus ⨁_{μ < λ} V_{μ}$ , where every weight $μ$ of $V$ has the form $μ = λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \in Z_{\geq 0}$ , and $V_{λ} = k v$ has dimension one; (c) $λ$ is dominant; (d) the sum of all proper $G$ -stable subspaces of $V$ is proper, so $V$ has a largest proper $G$ -submodule and the quotient by it is a simple $G$ -module generated by the image of $v$ .

Facts & Assumptions

Given: A split reductive group $(G, T)$ with Borel $B \supseteq T$ , unipotent radical $U = B_{u}$ , opposite unipotent group $U^{-}$ , a rational representation $(V, r)$ generated as a $G$ -module by a primitive vector $v$ of weight $λ$ .

[F1]

Big cell. The multiplication map $U^{-} \times B \to G$ is an open immersion onto a dense open subscheme of $G$ (Bruhat decomposition for a split reductive group).

[F2]

Morphisms equal on a dense open. Since the split reductive group $G$ is smooth, hence reduced, two morphisms from $G$ to a separated target that agree on a dense open subscheme are equal (Agreement on a schematically dense open).

[F3]

Root expansion. For $α \in Φ$ and a weight vector $w \in V_{μ}$ one has $u_{α} (c) \cdot w = w + \sum_{i \geq 1} c^{i} w_{i}$ with $w_{i} \in V_{μ + i α}$ , only finitely many nonzero (Expansion of a root-group translate of a weight vector).

[F4]

Primitive vectors. $v \neq 0$ is fixed by $U$ and satisfies $t \cdot v = λ (t) v$ , and $B$ acts on the line $k v$ through $λ$ (Primitive vectors for a Borel pair).

[F5]

Weights and the order. The weight spaces are the eigenspaces of $T$ , the order on $X$ is generated by the positive simple roots, and an expression of an element of the root lattice as $\sum_{α \in Δ} m_{α} α$ is unique (Weights, dominant weights and the highest-weight order of a rational representation, Combinatorics of a reduced root datum).

[F6]

Simple reflections. For a simple root $α$ the reflection acts on weights by $s_{α} (μ) = μ - ⟨ μ, α^{\lor} ⟩ α$ , and it is realized by an element of the normalizer of $T$ (The normalizer of the torus permutes weight spaces, The Weyl group, Borel subgroups and chambers, Structure of SL_2 and root coordinates).

[F7]

Every finite subset of a rational representation lies in a finite-dimensional subrepresentation. (Every element of a comodule lies in a finite-dimensional subcomodule)

[F8]

Ordered products of the negative root groups give $U^{-}$ . (Root subgroups of a split reductive group)

Proof

technique · direct

1.1F1F2F4F7givenalgebra

(a). By [F7], a finite-dimensional $G$ -submodule contains $v$ ; since $v$ generates $V$ , that submodule is $V$ . Let $Z$ be the $U^{-}$ -submodule generated by $v$ , defined as the span of the coefficients of its $U^{-}$ -coaction. Coassociativity makes this span a subcomodule; equivalently it is the smallest subcomodule containing $v$ . If a linear functional $f \in V^{*}$ vanishes on $Z$ , the regular function $g \mapsto f (g v)$ restricts to zero on $U^{-} B$ : universally over every base algebra, $u b v$ is the scalar by which $b$ acts on $k v$ times $u v$ , and the latter has all its coefficients in $Z$ . By [F1] and [F2] this regular function is zero on $G$ . Thus $f$ vanishes on the coefficients of the $G$ -coaction of $v$ , whose span is $V$ . In finite dimension the annihilator of $Z$ being zero implies $Z = V$ , proving (a). No vector over a general base algebra is treated as an element of $V$ .

2.1F3F4F5F8step 1.1

(b), weights. By (a), $V$ is the coefficient span of the $U^{-}$ -coaction of $v$ . The ordered product of the negative root groups is $U^{-}$ by the root-group supplier. Applying [F3] successively to this universal product shows that all these coefficients lie in $k v + \sum_{μ < λ} V_{μ}$ : each application of $u_{- α} (c)$ replaces a weight $μ$ by $μ$ or by $μ - i α$ with $i \geq 1$ , and $λ - \sum m_{α} α$ with $m_{α} \geq 0$ is the corresponding form, the simple roots generating the positive roots. Distinct weights give independent eigenspaces, so the sum $k v \oplus ⨁_{μ < λ} V_{μ}$ is direct, and the $V_{λ}$ -component of every element of $V$ is a multiple of $v$ ; since $v \neq 0$ this gives $V = k v \oplus ⨁_{μ < λ} V_{μ}$ and $V_{λ} = k v$ of dimension one.

3.1F5F6step 2.1

(c). Let $α \in Δ$ be a simple root and let $n$ represent the reflection $s_{α}$ . Since $v$ has weight $λ$ , the vector $n \cdot v$ has weight $s_{α} (λ)$ by [F6]; it is nonzero, so $s_{α} (λ)$ is a weight of $V$ , hence by step 2.1 of the form $λ - \sum_{β \in Δ} m_{β} β$ with $m_{β} \geq 0$ . Writing $s_{α} (λ) = λ - ⟨ λ, α^{\lor} ⟩ α$ and comparing the expansions of $s_{α} (λ) - λ$ in the basis $Δ$ of the root lattice, uniqueness of the coefficients [F5] gives $⟨ λ, α^{\lor} ⟩ = m_{α} \geq 0$ . Hence $λ$ is dominant.

3.2F5step 1.1step 2.1

(d). Every proper $G$ -stable subspace $W ⊊ V$ has $W \cap V_{λ} = 0$ : if $W$ contained a nonzero element of $V_{λ} = k v$ , then it would contain $v$ , hence by (a) the whole $U^{-}$ -span of $v$ , which is $V$ , a contradiction. Since $W$ is $T$ -stable, its weight components lie in $W$ : applying the coordinate functional of each character basis element to its coaction extracts that component in $W$ . Thus the vanishing of $W \cap V_{λ}$ forces every proper $G$ -stable subspace to be contained in $⨁_{μ < λ} V_{μ}$ , and so is their sum $M$ ; since $v \notin M$ the sum is proper and is therefore the largest proper $G$ -stable subspace. For any nonzero $G$ -submodule $Q \subseteq V / M$ , its preimage in $V$ is a $G$ -stable subspace strictly containing $M$ , hence equal to $V$ ; so $V / M$ is simple, and it is generated by the image of $v$ .

4.1step 1.1step 2.1step 3.1step 3.2∎

Steps 1.1, 2.1, 3.1 and 3.2 prove (a), (b), (c) and (d).

Remarks

The density step in (a) is the scheme-theoretic form of Milne's "as $U^{-} B$ is dense in $G$ , $V$ is spanned by $U^{-} B v$ "; the hypothesis that one vector generates the rational module already forces that module to be finite-dimensional, by the finite-subcomodule theorem.
The weights of $V$ below $λ$ are exactly the elements of the form $λ - \sum_{α \in Δ} m_{α} α$ that occur; the order is the dominance order generated by the positive simple roots.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Primitive vectors from standard maximal parabolics

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(G, T)$ be a split reductive group over $k$ with Borel $B \supseteq T$ , base $Δ = {α_{i}}_{i \in I}$ and fundamental weights $ω_{i}$ (Weights, dominant weights and the highest-weight order of a rational representation, The root datum of a split reductive group). For every $i \in I$ there exist a finite-dimensional rational representation $V_{i}$ of $G$ and a primitive vector $v_{i} \in V_{i}$ whose weight $λ_{i}$ satisfies $⟨ λ_{i}, α_{j}^{\lor} ⟩ = 0 (j \neq i), ⟨ λ_{i}, α_{i}^{\lor} ⟩ > 0.$ When the root datum is semisimple, $λ_{i} = ⟨ λ_{i}, α_{i}^{\lor} ⟩ ω_{i}$ .

Facts & Assumptions

Given: A split reductive group $(G, T)$ with Borel $B \supseteq T$ , base $Δ = {α_{i}}_{i \in I}$ and $i \in I$ .

[F1]

Standard maximal parabolic. $P_{i} = P_{Δ ∖ {α_{i}}}$ is the standard parabolic subgroup containing $B$ whose Levi subgroup contains $T$ and the root groups $U_{\pm α_{j}}$ for $j \neq i$ , and $P_{i}$ does not contain $U_{- α_{i}}$ (Parabolic subgroups and Levi decomposition, Standard Levi subgroups of a split reductive group, Parabolic subgroups of an affine algebraic group, Root subgroups of a split reductive group).

[F2]

Chevalley's theorem. There is a finite-dimensional rational representation $V$ of $G$ with a line $L \subseteq V$ such that ${Stab}_{G} (L) = P_{i}$ scheme-theoretically (Chevalley: every closed subgroup is a line stabilizer).

[F3]

Generated modules. If $V$ is generated as a $G$ -module by a primitive vector of weight $λ$ , then $λ$ is dominant, the weights of $V$ below $λ$ are of the form $λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ , and $λ$ has multiplicity one (Modules generated by a primitive vector, Primitive vectors for a Borel pair).

[F4]

Normalizer action and reflections. For a representative $n$ of a Weyl group element $w$ , the translate of a weight vector $v \in V_{μ}$ by $n$ has weight $w μ$ ; the simple reflection acts by $s_{α_{j}} (μ) = μ - ⟨ μ, α_{j}^{\lor} ⟩ α_{j}$ (The normalizer of the torus permutes weight spaces, The Weyl group, Borel subgroups and chambers).

[F5]

Coefficient uniqueness. An element of $Z Φ$ has a unique expression as $\sum_{α \in Δ} m_{α} α$ with $m_{α} \in Z$ (Combinatorics of a reduced root datum).

[F6]

Semisimple root datum. If the root datum is semisimple, then $X_{0} = {x \in X : ⟨ x, α^{\lor} ⟩ = 0 \forall α} = 0$ and the fundamental weights form a $Q$ -basis of $X \otimes Q$ (Combinatorics of a reduced root datum, The root datum of a split reductive group).

Proof

technique · direct

1.1F2F3given

Let $(V, L)$ be as in [F2] and let $v \neq 0$ generate $L$ . Since $P_{i} \supseteq B \supseteq U$ and $P_{i} \supseteq T$ , the line $L$ is $B$ -stable, so $v$ is primitive; let $λ_{i}$ be its weight. Replacing $V$ by the $G$ -submodule generated by $v$ does not change the line $L$ , and by [F3] the weight $λ_{i}$ is then dominant.

2.1F1F4step 1.1

For $j \neq i$ choose a representative $n \in P_{i} (k)$ of the simple reflection $s_{α_{j}}$ , which exists because $P_{i}$ contains $U_{\pm α_{j}}$ and $T$ by [F1]. Since $n$ stabilizes $L$ , the vector $n \cdot v$ is a nonzero multiple of $v$ , hence has weight $λ_{i}$ ; on the other hand by [F4] it has weight $s_{α_{j}} (λ_{i}) = λ_{i} - ⟨ λ_{i}, α_{j}^{\lor} ⟩ α_{j}$ . Therefore $⟨ λ_{i}, α_{j}^{\lor} ⟩ α_{j} = 0$ and $⟨ λ_{i}, α_{j}^{\lor} ⟩ = 0$ .

2.2F1F3F4step 1.1

For the remaining index, suppose $⟨ λ_{i}, α_{i}^{\lor} ⟩ = 0$ . Then $s_{α_{i}}$ fixes $λ_{i}$ ; choose a representative $n_{i} \in N_{G} (T) (k)$ of this simple reflection. It takes $v$ to a nonzero vector of weight $s_{α_{i}} (λ_{i}) = λ_{i}$ . By [F3], the weight- $λ_{i}$ space in the module generated by $v$ is one-dimensional, so $n_{i} v$ is a nonzero multiple of $v$ and $n_{i}$ stabilizes $L$ . Since ${Stab}_{G} (L) = P_{i}$ , this gives $n_{i} \in P_{i}$ . But $U_{α_{i}} \subseteq B \subseteq P_{i}$ , and conjugation by $n_{i}$ carries $U_{α_{i}}$ onto $U_{- α_{i}}$ ; this contradicts [F1], which says $U_{- α_{i}} \subseteq̸ P_{i}$ . Thus $⟨ λ_{i}, α_{i}^{\lor} ⟩ \neq 0$ , and dominance gives $⟨ λ_{i}, α_{i}^{\lor} ⟩ > 0$ .

3.1F5F6step 2.1step 2.2

If the root datum is semisimple, then by [F6] the element $λ_{i}$ has the unique expression $\sum_{j} ⟨ λ_{i}, α_{j}^{\lor} ⟩ ω_{j}$ in the basis of fundamental weights; by steps 2.1 and 2.2 only the $i$ -th coefficient is nonzero, so $λ_{i} = ⟨ λ_{i}, α_{i}^{\lor} ⟩ ω_{i}$ with $⟨ λ_{i}, α_{i}^{\lor} ⟩ > 0$ .

4.1step 1.1step 2.2step 3.1∎

Steps 1.1 to 2.1 produce, for each $i \in I$ , a finite-dimensional representation and a primitive vector with the required pairings, and the stated semisimple form of the weight.

Remarks

The proof uses Chevalley's line-stabilizer theorem to produce the maximal parabolic as a stabilizer, and then the Weyl-group action (the normalizer lemma) to read off the simple-coroot pairings; this is the intrinsic form of the Mostow argument recorded in the source's NOTES.
For a general reductive root datum the element $λ_{i}$ need not be a multiple of a fundamental weight, because the annihilator of the coroots can be nonzero; only the pairings are asserted.

PropositionStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Primitive vectors of the induced coordinate module

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $E (λ)$ be the induced coordinate module of a split reductive group $(G, T)$ with Borel $B$ and unipotent radical $U = B_{u}$ (The induced coordinate module E(lambda)). If $E (λ) \neq 0$ , then the space $E (λ)^{U}$ of $U$ -fixed elements is one-dimensional, its nonzero elements are primitive vectors of weight $λ$ (Primitive vectors for a Borel pair), and evaluation at the identity $f \mapsto f (1)$ is an isomorphism $E (λ)^{U} \to k$ . In particular $E (λ) \neq 0$ if and only if $E (λ)$ contains a primitive vector of weight $λ$ , unique up to scalar.

Facts & Assumptions

Given: A split reductive group $(G, T)$ with Borel $B \supseteq T$ , unipotent radical $U = B_{u}$ , opposite Borel $B^{0} = B^{-}$ , and an element $λ \in X (T)$ with $E (λ) \subseteq O (G)$ as in the cited definition.

[F1]

Big cell. $U \times B^{0} \to G$ , $(u, b) \mapsto u b$ , is an open immersion onto a dense open subscheme of $G$ , and $G$ is smooth, hence reduced (Bruhat decomposition for a split reductive group).

[F2]

Determination on the big cell. Two elements of $E (λ)$ agreeing on the image of $U \times B^{0}$ agree on $G$ : morphisms from the reduced scheme $G$ equal on a dense open are equal (Agreement on a schematically dense open).

[F3]

The action on $U$ . For $f \in E (λ)$ put $f_{U} (u) = f (u^{- 1})$ . For $u_{0} \in U (R)$ one has $(u_{0} f)_{U} (u) = f ((u u_{0})^{- 1}) = f_{U} (u u_{0})$ ; in particular $f \in E (λ)^{U}$ is fixed by every $U (R)$ exactly when $f_{U}$ is invariant under right translation by $U (R)$ (The induced coordinate module E(lambda), Rational representations of an affine group scheme are comodules of its coordinate Hopf algebra).

[F4]

Unipotent fixed vectors. A nonzero rational representation of the unipotent group $U$ has a nonzero $U$ -fixed vector; and a $U$ -invariant regular function on $U$ is constant, since right-translation invariance gives $f_{U} (u) = f_{U} (e)$ by translating the identity by $u$ (Unipotent algebraic groups and unipotent representations).

[F5]

Translation law. $f (u b) = f (u) λ (b^{- 1})$ for $u \in U (R)$ , $b \in B^{0} (R)$ , and $f (t) = f (1) λ (t^{- 1})$ for $t \in T (R)$ (The induced coordinate module E(lambda)).

Proof

technique · direct

1.1F1F2F5given

If $f \in E (λ)$ vanishes on $U$ , then $f = 0$ : by [F5] $f$ vanishes on the whole big cell $U \cdot B^{0}$ [F1], and [F2] applies.

1.2F4given

If $E (λ) \neq 0$ , then $E (λ)^{U} \neq 0$ : $E (λ)$ is a nonzero rational representation of the unipotent group $U$ , so it has a nonzero fixed vector by [F4].

2.1F3F4step 1.1step 1.2

For $f \in E (λ)^{U}$ the function $f_{U}$ is invariant under right translation by $U (R)$ for every $R$ by [F3], hence constant by [F4]; its constant value is $f_{U} (1) = f (1)$ , so $f$ is determined by the scalar $f (1)$ . Consequently the $k$ -linear evaluation map $E (λ)^{U} \to k$ , $f \mapsto f (1)$ , is injective; by step 1.2 it is nonzero (a nonzero fixed vector has $f (1) \neq 0$ by step 1.1), so $E (λ)^{U}$ is one-dimensional and evaluation is an isomorphism onto $k$ .

3.1F5step 2.1

Let $0 \neq f \in E (λ)^{U}$ . For $t \in T (R)$ one has $(t f) (1) = f (t^{- 1}) = f (1) λ (t)$ by [F5], so $t \cdot f = λ (t) f$ and $f$ is a $T$ -eigenvector of weight $λ$ . Since $f$ is $U$ -fixed, it is primitive of weight $λ$ by the definition.

4.1step 1.2step 2.1step 3.1∎

Conversely, a primitive vector of weight $λ$ in $E (λ)$ is $U$ -fixed, hence lies in the one-dimensional space $E (λ)^{U}$ of step 2.1; so it is unique up to a nonzero scalar, and its existence forces $E (λ) \neq 0$ . Together with steps 1.2, 2.1 and 3.1 this proves all assertions.

Remarks

The statement is the source's Proposition 22.22 in the conventions of this page: the fixed space is $E (λ)^{U}$ for the unipotent radical of the Borel $B$ used to define primitive vectors, and the big cell is $U \cdot B^{0} = U \cdot B^{-}$ . The earlier scaffold's formulation with $E (λ)^{U^{-}}$ was corrected here; the computations for ${SL}_{2}$ in the example show that $E (λ)^{U^{-}}$ is not the one-dimensional space of primitive vectors.
The one-dimensionality of $E (λ)^{U}$ is what makes the primitive vector "unique up to scalar" and underlies the classification.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Simple rational representations have a unique highest weight

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(V, r)$ be a simple rational representation of a split reductive group $(G, T)$ with Borel subgroup $B \supseteq T$ (Borel subgroups, maximal tori and Borel pairs, Root subgroups of a split reductive group). Then: (a) $V$ contains a primitive vector $v$ , and it is unique up to multiplication by a nonzero scalar; (b) its weight $λ$ is dominant and the weight space $V_{λ} = k v$ is one-dimensional; (c) every weight $μ$ of $V$ satisfies $μ \leq λ$ , i.e. $μ = λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ ; (d) any two primitive vectors of $V$ have the same weight; this weight is called the highest weight of $V$ .

Facts & Assumptions

Given: A simple rational representation $(V, r)$ of the split reductive group $(G, T)$ with Borel $B \supseteq T$ , unipotent radical $U = B_{u}$ and $B / U ≅ T$ .

[F1]

$B$ is trigonalizable. The split-Borel root-subgroup theorem gives the multiplication map $\prod_{α \in Φ^{+}} U_{α} \to U = B_{u}$ as a $T$ -equivariant isomorphism of varieties for any ordering (Root subgroups of a split reductive group). Since $B$ is split connected solvable with unipotent radical $U$ and $B / U ≅ T$ diagonalizable, $B$ is trigonalizable (Trigonalizable algebraic groups, Unipotent algebraic groups and unipotent representations, Borel subgroups, maximal tori and Borel pairs).

[F2]

Invariant flags. Every finite-dimensional rational representation of a trigonalizable group has a complete flag of subrepresentations (Trigonalizable groups, invariant flags and embeddings into T_n).

[F3]

Finite dimensionality. Simple rational representations of an affine group scheme of finite type are finite-dimensional (Simple rational representations are finite-dimensional).

[F4]

Modules generated by a primitive vector. If a rational representation $W$ of $(G, T)$ is generated as a $G$ -module by a primitive vector of weight $λ$ , then $λ$ is dominant, $W_{λ}$ is one-dimensional, and every weight of $W$ is of the form $λ - \sum_{α \in Δ} m_{α} α$ , $m_{α} \geq 0$ (Modules generated by a primitive vector, Primitive vectors for a Borel pair).

[F5]

The simple roots form a basis of the root lattice, so an expansion in $Δ$ has unique coefficients, and a sum of nonnegative multiples of simple roots equalling zero has all coefficients zero. (Simple roots form a signed integral basis)

Proof

technique · direct

1.1F1F2F3given

By [F1] and [F2], and the finite-dimensionality of $V$ from [F3], the $B$ -module $V$ has a complete flag of subrepresentations; its one-dimensional member is a $B$ -stable line $k v$ with $v \neq 0$ , so $v$ is primitive.

2.1F4step 1.1

Since $V$ is simple and $v \neq 0$ , the $G$ -submodule generated by $v$ is nonzero, hence equal to $V$ ; so $V$ is generated by the primitive vector $v$ . Let $λ$ be its weight. By [F4] the weight $λ$ is dominant, $V_{λ} = k v$ is one-dimensional, and every weight of $V$ has the form $λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ . This proves (b) and (c).

3.1F4F5step 2.1

Let $v^{'}$ be another primitive vector of $V$ , of weight $λ^{'}$ . Since $V$ is simple, $v^{'}$ also generates $V$ as a $G$ -module, so [F4] applied to $v^{'}$ gives $λ = λ^{'} - \sum_{α \in Δ} m_{α}^{'} α$ with $m_{α}^{'} \geq 0$ , while [F4] applied to $v$ gives $λ^{'} = λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ . Adding gives $0 = \sum_{α \in Δ} (m_{α} + m_{α}^{'}) α$ , so by [F5] all $m_{α} = m_{α}^{'} = 0$ and $λ = λ^{'}$ .

4.1step 1.1step 2.1step 3.1∎

Any two primitive vectors $v, v^{'}$ have the same weight $λ$ by step 3.1, and since $V_{λ} = k v$ is one-dimensional by step 2.1, $v^{'}$ is a nonzero scalar multiple of $v$ ; this proves (a) and (d). Together with step 2.1 the four assertions hold, and the common weight is the highest weight of $V$ .

Remarks

The primitive vector exists because a split connected solvable group is trigonalizable, so the invariant-flag theorem applies to the finite-dimensional module $V$ .
Uniqueness up to scalar follows from the two inclusions $λ^{'} \leq λ$ and $λ \leq λ^{'}$ together with the one-dimensionality of the top weight space.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Central characters and descent along a central isogeny

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(G, T)$ be a split semisimple group over $k$ with root system $Φ$ and root lattice $Q = Z Φ$ (The root datum of a split reductive group). Then: (a) the scheme-theoretic centre $Z (G)$ equals $⋂_{α \in Φ} \ker α$ , and restriction of characters identifies $X (Z (G))$ with $X (T) / Q$ (Centre, radical and semisimple quotient of a reductive group, Split diagonalizable groups are dual to abelian groups); (b) if $V$ is a simple rational representation of $G$ of highest weight $λ$ , then $Z (G)$ acts on $V$ through the character $λ + Q \in X (Z (G))$ (Simple rational representations have a unique highest weight). Now let $π : (G^{'}, T^{'}) \to (G, T)$ be a central isogeny of split reductive groups, so that $X (T) \subseteq X (T^{'})$ is a subgroup of finite index. Let $λ \in X (T^{'})$ and let $V^{'}$ be a simple $G^{'}$ -module of highest weight $λ$ ; then $V^{'}$ factors through $G$ if and only if $λ \in X (T)$ , and the descended $G$ -module is simple. For identification of highest weights, choose Borel subgroups $B^{'} \supseteq T^{'}$ and $B \supseteq T$ with $π (B^{'}) = B$ ; relative to these compatible source and target Borel pairs, the descended module has highest weight $λ$ . Equivalently, for the finite central subgroup scheme $N = \ker π$ (Quotient sheaves and representable quotients for pre-relations and group actions), a simple $G^{'}$ -module whose highest weight lies in $X (T)$ has $N$ acting trivially and hence descends along the fppf quotient $G^{'} / N = G$ (Affine finite locally free equivalence relations have finite locally free scheme quotients).

Facts & Assumptions

Given: AC; a split semisimple group $(G, T)$ with root system $Φ$ and root lattice $Q = Z Φ$ ; a simple rational representation $(V, r)$ of $G$ of highest weight $λ$ ; and a central isogeny $π : (G^{'}, T^{'}) \to (G, T)$ of split reductive groups with kernel $N$ . For highest-weight identification choose $B^{'} \supseteq T^{'}$ and $B \supseteq T$ with $π (B^{'}) = B$ .

[F1]

Centre inside the torus. For every reductive algebraic group $G$ and maximal torus $T$ one has $Z (G) \subseteq T$ , and $Z (G)$ is of multiplicative type (Centre, radical and semisimple quotient of a reductive group, Groups of multiplicative type and tori).

[F2]

Root groups generate, and the conjugation formula. $G$ is generated by $T$ and the root groups $U_{α}$ ( $α \in Φ$ ), each $u_{α} : G_{a} \to U_{α}$ is an isomorphism of group schemes. A smooth $T$ -stable subgroup contains $U_{α}$ precisely when its Lie algebra contains $g_{α}$ . Also, $t u_{α} (c) t^{- 1} = u_{α} (α (t) c)$ for all $t \in T (R)$ , $c \in R$ , as derived universally over the torus coordinate domain in [F3] of Expansion of a root-group translate of a weight vector (Root subgroups of a split reductive group, Roots and root groups of a split reductive group).

[F3]

Weights of a simple representation. If $W$ is a simple rational representation of a split reductive group $(H, S)$ of highest weight $ν$ , then $ν$ is dominant, $W_{ν}$ is one-dimensional, and every weight of $W$ has the form $ν - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ (Simple rational representations have a unique highest weight).

[F4]

Diagonalizable duality. For split diagonalizable groups, $X$ is an exact contravariant equivalence with finitely generated abelian groups (Split diagonalizable groups are dual to abelian groups). Consequently, for a morphism $ψ : T \to H$ of split diagonalizable groups with kernel $K$ , one has $X (K) = coker (ψ^{*} : X (H) \to X (T))$ , the character lattice of $K$ being identified with the quotient of $X (T)$ by the image of $ψ^{*}$ (Character and cocharacter lattices of a split torus).

[F5]

Quotients and their universal property. The fppf quotient of an affine finite-type group scheme by a finite locally free equivalence relation exists and represents the quotient sheaf, with the usual universal property: a morphism constant on the equivalence classes factors uniquely through the quotient (Quotient sheaves and representable quotients for pre-relations and group actions, Affine finite locally free equivalence relations have finite locally free scheme quotients).

[F6]

The kernel $N$ of the central isogeny is finite and lies in $Z (G^{'}) \subseteq T^{'}$ . Its restriction to the split tori is an isogeny $T^{'} \to T$ with kernel $N$ ; thus pullback identifies $X (T)$ with a finite-index subgroup of $X (T^{'})$ . This torus assertion does not require the total Lie differential to be an isomorphism. (Character and cocharacter lattices of a split torus, Split diagonalizable groups are dual to abelian groups)

[F7]

A subgroup scheme that is both unipotent and diagonalizable is trivial. The scheme image of a group homomorphism is its exact kernel quotient; in particular a homomorphism with trivial scheme kernel is an isomorphism onto its closed image. (A subgroup that is both unipotent and diagonalizable is trivial, Group images are exact kernel quotients and preserve affine smooth connected properties)

[F8]

The root big cell $U^{-} \times T \times U \to G$ is an open immersion with dense image; $G$ is smooth and geometrically connected, hence geometrically integral. (Bruhat decomposition for a split reductive group, Split reductive groups)

Proof

Proof technique: direct.

1.1F1F2given

For (a), first inclusion: let $R$ be a $k$ -algebra and let $z \in Z (G) (R)$ . By [F1] $z \in T (R)$ . Since $z$ is central, $z u_{α} (c) z^{- 1} = u_{α} (c)$ for every $α \in Φ$ and every $c \in R$ , while the conjugation formula of [F2] gives $z u_{α} (c) z^{- 1} = u_{α} (α (z) c)$ . As $u_{α}$ is an isomorphism of group schemes, $u_{α} (R)$ is injective, so $α (z) c = c$ for all $c$ ; setting $c = 1$ gives $α (z) = 1$ . Hence $z \in (⋂_{α \in Φ} \ker α) (R)$ and $Z (G) \subseteq ⋂_{α \in Φ} \ker α$ .

1.2F2F8algebra

Conversely let $t \in (⋂_{α} \ker α) (R)$ . By [F2], conjugation by $t$ is the identity on $T_{R}$ and each $U_{α, R}$ , so it is the identity on the base-changed root big cell of [F8]. Restriction $O (G) \to O (U^{-} \times T \times U)$ is injective because this is a dense open in an integral affine scheme. Tensoring that injection with the $k$ -vector space $R$ preserves injectivity: each finite tensor expression can be checked in a finite-dimensional subspace of $R$ , where the map is a finite direct sum of the original injection. Thus restriction remains injective even for nonreduced $R$ , and the two endomorphisms $c_{t}$ and $id$ of $G_{R}$ have equal maps on coordinate rings. Hence $t$ is central. This proves the scheme equality in (a); the same argument works for split reductive groups.

1.3F5F6

The restriction $π ∣_{T^{'}} : T^{'} \to T$ is, by [F6], a finite locally free surjection of split tori with kernel $N$ , so it identifies $T$ with the fppf quotient $T^{'} / N$ [F5]. Hence a character $χ \in X (T^{'})$ vanishes on $N$ if and only if it factors through $T^{'} / N = T$ , i.e. if and only if $χ \in X (T) \subseteq X (T^{'})$ : a character trivial on $N$ is constant on $N$ -orbits and factors through the quotient by its universal property, while a character pulled back from $T$ is trivial on $N$ .

2.1F2F6F7step 1.3algebra

The root groups, rather than the total Lie differential, give the root correspondence. For $α^{'} \in Φ^{'}$ , the kernel of $π ∣_{U_{α^{'}}}$ is $N \cap U_{α^{'}}$ . It is unipotent as a subgroup of $U_{α^{'}} ≃ G_{a}$ and diagonalizable as a subgroup of $T^{'}$ , so it is trivial by [F7]. The restriction is therefore an isomorphism onto a smooth connected closed image $H ≃ G_{a}$ . Centrality of $N$ and the root conjugation formula imply $α^{'} ∣_{N} = 1$ , so step 1.3 gives a unique character $α \in X (T)$ whose pullback is $α^{'}$ . The image $H$ is $T$ -stable and its nonzero tangent line has weight $α$ , so $α$ is a root of $G$ . The root-subgroup containment criterion of [F2] then gives $U_{α} \subseteq H$ ; both are smooth connected curves, hence $H = U_{α}$ . Distinct $α^{'}$ give distinct $α$ by injectivity of torus pullback. Since the isogeny preserves dimension and torus rank, and $\dim G = \dim T + ∣ Φ ∣$ , this injection of roots is bijective. For compatible Borels, it identifies the positive roots: the image of $B^{'}$ is the Borel generated by $T$ and the images of its positive root groups. Rank-one normalizer representatives map to the corresponding target reflections, so torus pullback intertwines the two reflections. Comparing their formulas gives $⟨ π^{*} x, α^{'}^{\lor} ⟩ = ⟨ x, α^{\lor} ⟩$ for every $x \in X (T)$ , and hence $π_{*} α^{'}^{\lor} = α^{\lor}$ . This includes inseparable isogenies and nonreduced $N$ .

2.2F4step 1.1step 1.2

For the second assertion of (a), apply [F4] to the morphism $ψ = (α)_{α \in Φ} : T \to \prod_{α \in Φ} G_{m}$ . Its kernel is $⋂_{α} \ker α = Z (G)$ by steps 1.1 and 1.2, and its dual $ψ^{*} : ⨁_{α \in Φ} Z e_{α} \to X (T)$ sends $e_{α}$ to $α$ , so its image is exactly the root lattice $Q$ . Therefore $X (Z (G)) = coker ψ^{*} = X (T) / Q$ , and the restriction map $X (T) \to X (Z (G))$ is the quotient map with kernel $Q$ .

3.1F3step 1.1step 2.2

For (b): let $μ$ be a weight of $V$ . By [F3] applied to the split reductive group $(G, T)$ , one has $μ = λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ ; hence for $z \in Z (G) (R)$ we get $α (z) = 1$ for all roots $α$ by step 1.1, so $μ (z) = λ (z) \prod_{α} α (z)^{- m_{α}} = λ (z)$ . The action of $z$ on the spanning weight spaces $V_{μ} \subseteq V_{R}$ is therefore the scalar $λ (z)$ , so $z$ acts on $V_{R}$ as $λ (z) id$ . This is the character of $Z (G)$ obtained by restricting $λ \in X (T)$ , which under step 2.2 corresponds to $λ + Q \in X (T) / Q = X (Z (G))$ .

4.1F1F2F3step 1.1step 1.2step 2.2step 3.1

The argument of steps 1.1, 1.2, 2.2 and 3.1 used only that the group is split reductive (generation by $T$ and the root groups, the conjugation formula, the weight description of simple modules, and [F1]), so it applies verbatim to the split reductive group $(G^{'}, T^{'})$ : $Z (G^{'}) = ⋂_{α^{'} \in Φ^{'}} \ker α^{'}$ with $X (Z (G^{'})) = X (T^{'}) / Q^{'}$ , and $Z (G^{'})$ acts on a simple $G^{'}$ -module of highest weight $λ$ through the character $λ + Q^{'}$ .

5.1F3F5F6step 1.3step 2.1step 4.1∎

Now let $λ \in X (T^{'})$ and let $V^{'}$ be a simple $G^{'}$ -module of highest weight $λ$ . By step 4.1 the centre $Z (G^{'})$ acts on $V^{'}$ through $λ + Q^{'}$ ; since $N \subseteq Z (G^{'})$ by [F6] and $Q^{'}$ vanishes on $Z (G^{'})$ , the subgroup $N$ acts on $V^{'}$ through the character $λ ∣_{N}$ . Hence $N$ acts trivially if and only if $λ ∣_{N} = 1$ , which by step 1.3 is exactly $λ \in X (T)$ . If $N$ acts trivially, the $G^{'}$ -action factors through the fppf quotient $G^{'} / N = G$ [F5], and the resulting $G$ -module is simple because every $G$ -submodule is a $G^{'}$ -submodule; conversely, if $V^{'}$ factors through $G$ , then $N$ acts trivially, so $λ \in X (T)$ . For the compatible Borel pairs chosen above, it remains to identify the highest weight: by [F3] applied to $(G^{'}, T^{'})$ , the $T^{'}$ -weights of $V^{'}$ are $λ - \sum_{α^{'} \in Δ^{'}} m_{α^{'}} α^{'}$ with $m_{α^{'}} \geq 0$ and $λ$ has multiplicity one, and by the root correspondence of step 2.1 the roots $α^{'}$ are pullbacks of the roots $α$ of $(G, T)$ with the same positive systems; hence the $T$ -weights of the descended module are $λ - \sum_{α \in Δ} m_{α} α$ , again with $λ$ of multiplicity one. So the descended $G$ -module is simple with highest weight $λ$ .

Remarks

The two inclusions of steps 1.1 and 1.2 are Milne's Corollary 21.8; the computation of the root lattice is the exact sequence $0 \to Q \to X (T) \to X (Z (G)) \to 0$ dual to $Z (G) ↪ T \overset{(α)}{\to} \prod G_{m}$ .
Part (b) is Milne 22.11: all weights of $V$ are congruent to $λ$ modulo the root lattice, and $Z (G)$ is precisely the part of $T$ on which every root is trivial.
The descent statement is Milne 22.12; the hypothesis that $λ$ lies in $X (T)$ is exactly the condition that the central subgroup $N = \ker π$ acts trivially, so that the fppf quotient $G = G^{'} / N$ receives the action.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Multiples of the fundamental weights are primitive weights in the semisimple case

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(G, T)$ be a split semisimple group over $k$ with base $Δ$ , fundamental weights $ω_{i}$ , and let $i \in Δ$ (The root datum of a split reductive group, Weights, dominant weights and the highest-weight order of a rational representation). Then there exists $d > 0$ such that $d ω_{i} \in X (T)$ and $d ω_{i}$ is the weight of a primitive vector of some finite-dimensional rational representation of $G$ (Primitive vectors from standard maximal parabolics, Abstract root data and their Weyl groups).

Facts & Assumptions

Given: AC; a split semisimple group $(G, T)$ with base $Δ$ , an index $i \in Δ$ , and a split Borel $B \supseteq T$ .

[F1]

Primitive vectors with prescribed coroot pairings. For every index $j$ there exist a finite-dimensional rational representation $V_{j}$ of $G$ and a primitive vector $v_{j} \in V_{j}$ whose weight $λ_{j}$ satisfies $⟨ λ_{j}, α_{k}^{\lor} ⟩ = 0$ for $k \neq j$ and $⟨ λ_{j}, α_{j}^{\lor} ⟩ > 0$ ; when the root datum is semisimple, $λ_{j} = ⟨ λ_{j}, α_{j}^{\lor} ⟩ ω_{j}$ (Primitive vectors from standard maximal parabolics).

[F2]

Fundamental weights. The fundamental weights $ω_{j} \in X (T) \otimes_{Z} Q$ are dual to the simple coroots, $⟨ ω_{j}, α_{k}^{\lor} ⟩ = δ_{j k}$ , and every $x \in X (T) \otimes Q$ decomposes as $x = \sum_{j} ⟨ x, α_{j}^{\lor} ⟩ ω_{j} + x_{0}$ with $⟨ x_{0}, α_{j}^{\lor} ⟩ = 0$ for all $j$ (Weights, dominant weights and the highest-weight order of a rational representation).

[F3]

Semisimple case. $G$ is semisimple if and only if $Z Φ$ has finite index in $X (T)$ (The root datum of a split reductive group, Centre, radical and semisimple quotient of a reductive group); in that case $Φ$ spans $X (T) \otimes Q$ (Combinatorics of a reduced root datum), so the annihilator $X_{0} = {x \in X (T) : ⟨ x, α^{\lor} ⟩ = 0 for all α \in Φ}$ is zero, as is its rational counterpart (Milne 22.9).

Proof

Given: AC; a split semisimple group $(G, T)$ with base $Δ$ , an index $i \in Δ$ , and a split Borel $B \supseteq T$ .

Proof technique: direct.

1.1F3

Since $G$ is semisimple, its root datum is semisimple and the annihilator $X_{0}$ is zero by [F3].

2.1F1F2step 1.1

Apply [F1] with $j = i$ : there exist a finite-dimensional rational representation $V_{i}$ and a primitive vector $v_{i} \in V_{i}$ of weight $λ_{i}$ with $d : = ⟨ λ_{i}, α_{i}^{\lor} ⟩$ a positive integer. In the semisimple case [F1] gives $λ_{i} = d ω_{i}$ , with $ω_{i}$ as defined in [F2].

3.1step 2.1∎

Therefore $d > 0$ , $d ω_{i} = λ_{i} \in X (T)$ , and $d ω_{i}$ is the weight of the primitive vector $v_{i}$ in the finite-dimensional rational representation $V_{i}$ of $G$ .

Remarks

The proof is Milne's second proof of Theorem 22.20 in the semisimple case: the Mostow argument (Lemma 22.24) produces a primitive vector whose weight pairs trivially with all simple coroots but one, and semisimplicity of the root datum forces that weight to be a positive multiple of the corresponding fundamental weight.
For a general reductive root datum the coroot annihilator can be nonzero, so the weight produced by the parabolic construction need not be a multiple of $ω_{i}$ ; only its pairings are controlled, which is why the reductive case is handled separately through the product $Z (G)_{t} \times G_{d e r}$ .

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Simple modules with equal highest weight are isomorphic

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $V_{1}, V_{2}$ be simple rational representations of a split reductive group $(G, T)$ with the same highest weight $λ$ (Simple rational representations have a unique highest weight). Then $V_{1} ≅ V_{2}$ .

Facts & Assumptions

Given: AC; two simple rational representations $V_{1}, V_{2}$ of the split reductive group $(G, T)$ with common highest weight $λ$ .

[F1]

Primitive vectors of simple modules. Each simple $V_{i}$ contains a primitive vector $v_{i}$ whose weight is its highest weight $λ$ , unique up to multiplication by a nonzero scalar; moreover $V_{i}$ is generated as a $G$ -module by $v_{i}$ , because $V_{i}$ is simple and $v_{i} \neq 0$ (Simple rational representations have a unique highest weight, Primitive vectors for a Borel pair).

[F2]

Modules generated by a primitive vector. If a rational representation $W$ is generated as a $G$ -module by a primitive vector $w$ of weight $λ$ , then $W_{λ} = k w$ is one-dimensional and every weight of $W$ is of the form $λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ (Modules generated by a primitive vector).

[F3]

Primitivity is closed under sums of equal weight. A vector is primitive of weight $λ$ if and only if it is fixed by the unipotent radical $U$ of a Borel $B \supseteq T$ and is a $T$ -eigenvector of weight $λ$ ; hence $v_{1} + v_{2} \in V_{1} \oplus V_{2}$ is primitive of weight $λ$ (Primitive vectors for a Borel pair, Weights, dominant weights and the highest-weight order of a rational representation).

Proof

Given: AC; two simple rational representations $V_{1}, V_{2}$ of the split reductive group $(G, T)$ with common highest weight $λ$ .

Proof technique: direct.

1.1F1F3

By [F1] choose primitive vectors $v_{i} \in V_{i}$ of weight $λ$ ; then $v : = v_{1} + v_{2}$ is a nonzero primitive vector of weight $λ$ in $V_{1} \oplus V_{2}$ by [F3].

2.1F2step 1.1

Let $V \subseteq V_{1} \oplus V_{2}$ be the $G$ -submodule generated by $v$ . By [F2] applied to $V$ and $v$ , the weight space $V_{λ}$ equals the line $k v$ ; in particular the only elements of $V$ of weight $λ$ are the multiples of $v$ .

3.1F1step 2.1

The projection $φ : V \to V_{2}$ , $(x_{1}, x_{2}) \mapsto x_{2}$ , is a $G$ -homomorphism with $φ (v) = v_{2} \neq 0$ , so its image is a nonzero $G$ -submodule of the simple module $V_{2}$ ; hence $φ$ is surjective. Its kernel is $V \cap V_{1}$ , a $G$ -submodule of the simple module $V_{1}$ , so the kernel is either $0$ or $V_{1}$ . If $V_{1} \subseteq V$ , then $v_{1} \in V$ has weight $λ$ , so $v_{1} = c v$ for some scalar $c$ by step 2.1; applying $φ$ gives $0 = c v_{2}$ , whence $c = 0$ and $v_{1} = 0$ , a contradiction. Therefore the kernel is $0$ and $φ : V \to V_{2}$ is an isomorphism.

4.1step 3.1∎

The same argument with the projection onto the first factor shows that $V \to V_{1}$ is an isomorphism as well, so $V_{1} ≅ V ≅ V_{2}$ .

Remarks

This is Milne's Theorem 22.19; the proof uses only that a module generated by a primitive vector has a one-dimensional top weight space, so that the diagonal line $k (v_{1} + v_{2})$ meets neither summand.
Uniqueness of the highest weight together with the existence theorem for dominant weights yields the classification of the simple rational representations of a split reductive group.

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Every dominant weight of a split semisimple group is a primitive weight

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(G, T)$ be a split semisimple group over $k$ and let $λ \in X (T)$ be dominant (Weights, dominant weights and the highest-weight order of a rational representation). Then there exists a (possibly infinite-dimensional) rational representation of $G$ containing a primitive vector of weight $λ$ ; consequently $E (λ) \neq 0$ (The induced coordinate module E(lambda)).

Facts & Assumptions

Given: AC; a split semisimple group $(G, T)$ with Borel $B \supseteq T$ , opposite Borel $B^{0} = B^{-}$ , unipotent radical $U = B_{u}$ , root datum $(X (T), Φ, α \mapsto α^{\lor})$ with base $Δ$ , and a dominant $λ \in X (T)$ .

[F1]

The induced coordinate module. $E (μ) \subseteq O (G)$ is the space of regular functions with $f (g b) = f (g) μ (b^{- 1})$ for all $k$ -algebras $R$ , $g \in G (R)$ , $b \in B^{0} (R)$ ; it is a $G$ -submodule of the regular representation (The induced coordinate module E(lambda)).

[F2]

Fixed vectors of $E (μ)$ . If $E (μ) \neq 0$ , then the fixed space $E (μ)^{U}$ is one-dimensional, evaluation $f \mapsto f (1)$ is an isomorphism $E (μ)^{U} \to k$ , and every nonzero $f \in E (μ)^{U}$ is a primitive vector of weight $μ$ satisfying $f (u) = f (1)$ for $u \in U (R)$ and $f (u b) = f (1) μ (b^{- 1})$ for $b \in B^{0} (R)$ ; in particular $E (μ) \neq 0$ if and only if $E (μ)$ contains a primitive vector of weight $μ$ (Primitive vectors of the induced coordinate module).

[F3]

The big cell and determination on it. $U \times B^{0} \to G$ , $(u, b) \mapsto u b$ , is an open immersion onto a dense open subscheme of $G$ ; $G$ is smooth and connected, hence reduced, so two morphisms from $G$ (or from $G \times B^{0}$ ) to a separated scheme that agree on that dense open agree everywhere (Bruhat decomposition for a split reductive group, Agreement on a schematically dense open, Smooth morphism of schemes).

[F4]

The longest element and dominance. The longest element $w_{0}$ of the Weyl group satisfies $w_{0} (Φ^{+}) = - Φ^{+}$ , $w_{0}^{2} = 1$ , and $w_{0} (Δ) = - Δ$ ; the Weyl group acts on $X (T)$ preserving the pairing with coroots, so $(- w_{0} λ)$ is dominant whenever $λ$ is (Combinatorics of a reduced root datum, Abstract root data and their Weyl groups, The Weyl group, Borel subgroups and chambers).

[F5]

Fundamental weights and the semisimple case. For a semisimple root datum, $X_{0} = {x \in X (T) : ⟨ x, α^{\lor} ⟩ = 0 \forall α} = 0$ , and $λ = \sum_{i \in Δ} ⟨ λ, α_{i}^{\lor} ⟩ ω_{i}$ with integer coefficients (Combinatorics of a reduced root datum, Weights, dominant weights and the highest-weight order of a rational representation).

[F6]

Fundamental weights. For every $i \in Δ$ there exists $d_{i} > 0$ with $d_{i} ω_{i} \in X (T)$ the weight of a primitive vector of a finite-dimensional rational representation (Multiples of the fundamental weights are primitive weights in the semisimple case).

[F7]

Tensor products. If $v$ and $v^{'}$ are primitive vectors of weights $μ$ and $μ^{'}$ , then $v \otimes v^{'}$ is primitive of weight $μ + μ^{'}$ (Tensor products of primitive vectors).

[F8]

Power extension over a normal domain. $G$ is a smooth connected affine group, so its coordinate ring is a normal domain; if $f$ is a regular function on a dense open subscheme of $G$ with $f^{d} \in O (G)$ for some $d > 0$ , then $f \in O (G)$ (regular local rings are normal, Smooth morphism of schemes, Power extension over a normal affine domain).

[F9]

Contragredient representation. The dual $V^{*}$ of a finite-dimensional rational representation $V$ is a rational representation, and matrix coefficients of a finite-dimensional rational representation are regular functions on $G$ (Contragredient (dual) rational representation, Rational representations and comodules of an affine group scheme).

Proof

Proof technique: direct.

1.1F1F2F3

Observation (a): for $μ \in X (T)$ , one has $E (μ) \neq 0$ if and only if the morphism $f_{μ} : U \cdot B^{0} \to A^{1}$ , $u b \mapsto μ (b^{- 1})$ , extends to $G$ . If $E (μ) \neq 0$ , pick $0 \neq f \in E (μ)^{U}$ ; by [F2] $f (u) = f (1)$ and $f (u b) = f (1) μ (b^{- 1})$ on the big cell, so $f / f (1) \in O (G)$ extends $f_{μ}$ . Conversely, if $F \in O (G)$ extends $f_{μ}$ , then the morphisms $G \times B^{0} \to A^{1}$ , $(g, b) \mapsto F (g b)$ and $(g, b) \mapsto F (g) μ (b^{- 1})$ , agree on the dense open $(U \cdot B^{0}) \times B^{0}$ (for $g = u b_{1}$ one has $F (g b) = f_{μ} (u b_{1} b) = μ (b^{- 1}) μ (b_{1}^{- 1}) = μ (b^{- 1}) F (g)$ ), hence by [F3] they agree on $G \times B^{0}$ and $F \in E (μ)$ , so $E (μ) \neq 0$ .

1.2F1F9

Observation (b): if $μ$ is the weight of a primitive vector of a finite-dimensional rational representation, then $E (- w_{0} μ) \neq 0$ . Let $v \in V$ be such a primitive vector and let $P \in G (k)$ represent $w_{0}$ ; choose $f \in V^{*}$ with $f (P v) \neq 0$ and put $F (g) = f (g P v)$ , a regular function by [F9]. For $b \in B^{0} (R)$ one has $P^{- 1} b P \in B (R)$ , since $w_{0}$ carries the opposite Borel to $B$ . The action of $B$ on the primitive line $k v$ is through the character extending $μ$ from $T$ and trivial on $U$ , so $(P^{- 1} b P) v = (w_{0} μ) (b) v$ , with $(w_{0} μ) (b) = μ (P^{- 1} b P)$ under this extension (The normalizer of the torus permutes weight spaces). Therefore $F (g b) = f (g b P v) = f (g P (P^{- 1} b P) v) = (w_{0} μ) (b) F (g) = (- w_{0} μ) (b^{- 1}) F (g)$ . Hence $F \in E (- w_{0} μ)$ and $F (1) = f (P v) \neq 0$ , so $E (- w_{0} μ) \neq 0$ .

1.3F4F5F6F7

For the dominant character $λ$ : $- w_{0} λ$ is dominant by [F4], and by [F5] $λ = \sum_{i \in Δ} m_{i} ω_{i}$ with $m_{i} = ⟨ λ, α_{i}^{\lor} ⟩ \in Z_{\geq 0}$ . If $λ = 0$ , a nonzero vector of the trivial representation is primitive of weight $0$ . If $λ \neq 0$ , put $D = \prod_{i : m_{i} > 0} d_{i}$ , where $d_{i}$ is as in [F6]; then $D λ = \sum_{i : m_{i} > 0} (D m_{i} / d_{i}) (d_{i} ω_{i})$ is a nonnegative integral combination of primitive weights, so $D λ$ is again a primitive weight by [F7], being the weight of a tensor product of primitive vectors.

2.1F4step 1.2step 1.3

By step 1.3 applied to the dominant character $- w_{0} λ$ , there exists $e > 0$ such that $e (- w_{0} λ)$ is a primitive weight. Observation (b) of step 1.2 with $μ = e (- w_{0} λ)$ gives $E (- w_{0} μ) = E (e λ) \neq 0$ , because $w_{0}^{2} = 1$ .

3.1F3F8step 1.1step 1.3step 2.1

By observation (a) of step 1.1 applied to $e λ$ , the function $f_{e λ}$ extends to $G$ . On the big cell $f_{e λ} (u b) = (e λ) (b^{- 1}) = (f_{λ} (u b))^{e}$ , so $(f_{λ})^{e} \in O (G)$ ; since $G$ is a normal affine scheme and $U \cdot B^{0}$ is a dense open subscheme, [F8] gives $f_{λ} \in O (G)$ .

4.1F2step 1.1step 3.1∎

By observation (a) of step 1.1 applied to $λ$ , the extension of $f_{λ}$ gives $E (λ) \neq 0$ ; by [F2] $E (λ)$ contains a primitive vector of weight $λ$ . Thus $E (λ)$ , a rational representation of $G$ , contains a primitive vector of weight $λ$ , as required.

Remarks

This is Milne's Lemma 22.26; the two observations (a) and (b) are exactly the two paragraphs of its proof, and the passage from $f_{d λ}$ to $f_{λ}$ is Lemma 22.23 (power extension over the normal domain $O (G)$ ).
When $G$ is semisimple, $X_{0} = 0$ , so every dominant $λ$ is a nonnegative integral combination of fundamental weights; this is where semisimplicity is used, and it is the reason the reductive case needs the separate product decomposition $Z (G)_{t} \times G_{d e r}$ and the central isogeny.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Dominant characters of a torus times a split semisimple group are primitive weights

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $Z$ be a split torus over $k$ and let $G_{0}$ be a split semisimple group over $k$ ; put $G = Z \times G_{0}$ with the product Borel pair. Let $λ = λ_{Z} + λ_{0} \in X (Z) \oplus X (T_{0})$ be dominant for the product (Groups of multiplicative type and tori, Split reductive groups). Then there is a rational representation of $G$ containing a primitive vector of weight $λ$ ; if $λ_{0}$ is dominant this is obtained by tensoring the one-dimensional representation of $Z$ of weight $λ_{Z}$ with a representation of $G_{0}$ carrying a primitive vector of weight $λ_{0}$ (Every dominant weight of a split semisimple group is a primitive weight, Tensor products of primitive vectors).

Facts & Assumptions

Given: AC; a split torus $Z$ with character lattice $X (Z)$ , a split semisimple group $(G_{0}, T_{0})$ with Borel $B_{0} \supseteq T_{0}$ and unipotent radical $U_{0}$ , the product $G = Z \times G_{0}$ with maximal torus $T = Z \times T_{0}$ and Borel $B = Z \times B_{0}$ , and a dominant $λ = (λ_{Z}, λ_{0}) \in X (Z) \oplus X (T_{0}) = X (T)$ .

[F1]

Characters of a split torus are one-dimensional representations. For $χ \in X (Z)$ let $k_{χ}$ be the one-dimensional rational representation of $Z$ on which $Z$ acts through $χ$ ; every nonzero vector of $k_{χ}$ is a $T_{Z}$ -eigenvector of weight $χ$ . The character $χ : Z \to G_{m} = {GL}_{1}$ itself defines this action. For the product $G$ the unipotent radical is $U = 1 \times U_{0}$ (Groups of multiplicative type and tori, Representations of diagonalizable groups split into character eigenspaces, Rational representations and comodules of an affine group scheme).

[F2]

Root datum and dominance of the product. The root datum of $(G, T)$ is $(X (Z) \oplus X (T_{0}), {0} \times Φ_{0}, X (Z)^{\lor} \oplus X (T_{0})^{\lor}, {0} \times Φ_{0}^{\lor})$ , so $⟨ λ, α^{\lor} ⟩ = ⟨ λ_{0}, α_{0}^{\lor} ⟩$ for every root $α = (0, α_{0})$ ; hence $λ$ is dominant for $G$ if and only if $λ_{0}$ is dominant for $G_{0}$ (The root datum of a split reductive group, Weights, dominant weights and the highest-weight order of a rational representation, Borel subgroups, maximal tori and Borel pairs, Character and cocharacter lattices of a split torus).

[F3]

Primitive vectors of products. If $v$ is primitive of weight $μ$ and $v^{'}$ is primitive of weight $μ^{'}$ for the same split reductive group, then $v \otimes v^{'}$ is primitive of weight $μ + μ^{'}$ (Tensor products of primitive vectors, Primitive vectors for a Borel pair).

[F4]

Semisimple factor. Every dominant character $λ_{0}$ of the split semisimple group $G_{0}$ is the weight of a primitive vector of a rational representation of $G_{0}$ (Every dominant weight of a split semisimple group is a primitive weight).

Proof

Proof technique: direct.

1.1F1

In the product $G = Z \times G_{0}$ , regard the one-dimensional representation $k_{λ_{Z}}$ as a $G$ -module on which $Z$ acts through $λ_{Z}$ and the factor $G_{0}$ acts trivially. Its nonzero vectors are fixed by $U = 1 \times U_{0}$ and are $T$ -eigenvectors of weight $(λ_{Z}, 0)$ , so each nonzero vector of $k_{λ_{Z}}$ is primitive of weight $(λ_{Z}, 0)$ for the pair $(B, T)$ .

1.2F2F4

By dominance of $λ$ and [F2], the character $λ_{0}$ is dominant for $G_{0}$ ; by [F4] there exist a rational representation $W$ of $G_{0}$ and a primitive vector $v_{0} \in W$ of weight $λ_{0}$ . Viewing $W$ as a $G$ -module through the projection $G \to G_{0}$ , the same $v_{0}$ is fixed by $U = 1 \times U_{0}$ and is a $T$ -eigenvector of weight $(0, λ_{0})$ , hence primitive of weight $(0, λ_{0})$ for $(B, T)$ .

2.1F1F2F3step 1.1step 1.2

By [F3] applied to the two primitive vectors of steps 1.1 and 1.2, the vector $1 \otimes v_{0}$ in the tensor product $k_{λ_{Z}} \otimes W$ is primitive of weight $(λ_{Z}, 0) + (0, λ_{0}) = λ$ for $(B, T)$ ; the tensor product is a rational representation of $G$ on which $(z, g_{0})$ acts by $λ_{Z} (z)$ on the first factor and through the $G_{0}$ -action on $W$ on the second.

3.1step 2.1∎

Thus $k_{λ_{Z}} \otimes W$ is a rational representation of the product $G = Z \times G_{0}$ containing the primitive vector $1 \otimes v_{0}$ of weight $λ$ , and it is obtained by tensoring the one-dimensional representation of $Z$ of weight $λ_{Z}$ with the representation $W$ of $G_{0}$ carrying the primitive vector $v_{0}$ of weight $λ_{0}$ .

Remarks

Dominance of $λ$ on the product is tested only on the simple coroots of the semisimple factor, because the roots of $Z \times G_{0}$ are the roots of $G_{0}$ pulled back along the projection; this is why the torus part is unrestricted, exactly as in Milne's reduction of Theorem 22.20 to the semisimple case.
The one-dimensional representation of the torus contributes a primitive vector of weight $(λ_{Z}, 0)$ , so tensor products with the semisimple part realize every dominant character of the product.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Every dominant character of a split reductive group is a highest weight

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(G, T)$ be a split reductive group over $k$ with Borel $B \supseteq T$ . Then every dominant $λ \in X (T)^{+}$ is the highest weight of a simple finite-dimensional rational representation of $G$ ; equivalently, there is a rational representation of $G$ containing a primitive vector of weight $λ$ (Weights, dominant weights and the highest-weight order of a rational representation).

Facts & Assumptions

Given: AC; a split reductive group $(G, T)$ with Borel $B \supseteq T$ , derived subgroup $G_{d e r}$ , and a dominant $λ \in X (T)$ .

[F1]

Decomposition of a reductive group. $Z (G)_{t}$ is a torus contained in $T$ , the derived subgroup $G_{d e r}$ is semisimple, $T_{d e r} = (T \cap G_{d e r})_{t}$ is a split maximal torus of $G_{d e r}$ , and the multiplication morphism $m : Z (G)_{t} \times G_{d e r} \to G$ is surjective with finite central kernel $N$ , so $m$ is a central isogeny; moreover every maximal torus of $G$ contains $Z (G)$ (Centre, radical and semisimple quotient of a reductive group, The derived subgroup, the derived series and solvable algebraic groups, Borel subgroups, maximal tori and Borel pairs, Maximal tori, field extensions, normal subgroups and derived groups, parts (d) and (e)). The maximal torus $T_{d e r} \subseteq T$ is split because a subtorus of a split torus has a character lattice that is a torsion-free quotient of the original lattice with trivial Galois action (Character and cocharacter lattices of a split torus). The product $H = Z (G)_{t} \times G_{d e r}$ is split reductive with maximal torus $T_{H} = Z (G)_{t} \times T_{d e r}$ and $m (T_{H}) = T$ (Split reductive groups).

[F2]

Dominant characters of the product. Let $Z$ be a split torus and let $T_{0}$ be a maximal torus of a split semisimple group $G_{0}$ . Every dominant character of $Z \times G_{0}$ is the weight of a primitive vector of a rational representation (Dominant characters of a torus times a split semisimple group are primitive weights).

[F3]

Descent along a central isogeny. Let $π : (H^{'}, T^{'}) \to (H^{''}, T^{''})$ be a central isogeny of split reductive groups with kernel $N^{'}$ , let $μ \in X (T^{'})$ and let $V^{'}$ be a simple $H^{'}$ -module of highest weight $μ$ . Then $V^{'}$ factors through $H^{''}$ if and only if $μ \in X (T^{''})$ ; if so, $N^{'}$ acts trivially and the descended $H^{''}$ -module is simple with highest weight $μ$ relative to compatible Borel pairs (Central characters and descent along a central isogeny).

[F4]

Modules generated by a primitive vector. If a rational representation of a split reductive group is generated by a primitive vector of weight $ν$ , then it has a largest proper submodule and the quotient by it is a simple module generated by the image of the vector, of highest weight $ν$ (Modules generated by a primitive vector).

[F5]

Simple modules. Every simple rational representation of $G$ contains a primitive vector whose weight is its highest weight, and every simple rational representation of the affine group scheme $G$ of finite type is finite-dimensional (Simple rational representations have a unique highest weight, Simple rational representations are finite-dimensional).

[F6]

Root groups, character descent and scheme images. The zero adjoint weight space is $Lie T$ (Roots and root groups of a split reductive group). Each root group is a smooth copy of $G_{a}$ with its root tangent weight; a smooth torus-stable subgroup contains that root group if its Lie algebra contains the corresponding root space. The rank-one normalizer represents the reflection $s_{α} (x) = x - ⟨ x, α^{\lor} ⟩ α$ , and positive root groups multiply to the Borel's unipotent radical (Root subgroups of a split reductive group). A subgroup scheme both unipotent and diagonalizable is trivial (A subgroup that is both unipotent and diagonalizable is trivial); a group homomorphism with trivial scheme kernel is an isomorphism onto its closed scheme image (Group images are exact kernel quotients and preserve affine smooth connected properties). The character anti-equivalence for split diagonalizable groups sends a scheme kernel to the cokernel of the character map; thus characters trivial on the kernel of a torus isogeny are precisely those pulled back from its target (Split diagonalizable groups are dual to abelian groups).

[F7]

Borels and positive systems. Borels containing a split maximal torus correspond bijectively to positive root systems and are the cocharacter subgroups for regular cocharacters (The Weyl group, Borel subgroups and chambers). The cocharacter decomposition gives $B = T ⋉ R_{u} (B)$ in this regular reductive case (Cocharacter limit subgroups).

Proof

Given: AC; a split reductive group $(G, T)$ with Borel $B \supseteq T$ , derived subgroup $G_{d e r}$ , and a dominant $λ \in X (T)$ .

Proof technique: direct.

1.1F1

By [F1] form $H = Z (G)_{t} \times G_{d e r}$ with maximal torus $T_{H} = Z (G)_{t} \times T_{d e r}$ and central isogeny $m : H \to G$ with finite kernel $N$ . Its torus restriction maps $T_{H}$ onto $T$ and induces an injection $m^{*} : X (T) ↪ X (T_{H})$ with finite cokernel. Put $λ_{H} = m^{*} λ$ , which lies in that image.

2.1F1F6step 1.1algebra

We establish the root correspondence, without assuming the total Lie differential is an isomorphism. The central kernel lies in $Z (H) \subseteq T_{H}$ by [F1] applied to $H$ . For a root $β$ of $H$ , $N \cap U_{β}$ is both unipotent and diagonalizable, so it is trivial; hence $m ∣_{U_{β}}$ is an isomorphism onto a smooth closed image $I_{β}$ by [F6]. Centrality makes $N$ act trivially on the root tangent line, so $β ∣_{N} = 1$ and exact character duality gives a unique $α \in X (T)$ with $m^{*} α = β$ . The image is $T$ -stable, with nonzero tangent weight $α$ , so $α$ is a root of $G$ , and the root-group containment criterion gives $U_{α} \subseteq I_{β}$ . Both are smooth connected curves, hence equal. This gives an injection of root sets; it is a bijection since $\dim H = \dim G$ , $\dim T_{H} = \dim T$ , and the root decomposition with one-dimensional root spaces gives $∣ Φ ∣ = \dim G - \dim T$ for either group.

3.1F6F7step 1.1step 2.1algebra

The torus isogeny maps the maximal subtorus of $\ker β$ onto that of $\ker α$ , so it maps the source rank-one centralizer into the target one. Hence a rank-one reflection representative $n_{β}$ maps into that target rank-one subgroup. Its action on $T$ is nontrivial, because torus pullback is injective and intertwines conjugation, so it represents $s_{α}$ by [F6]. Comparing the two reflection formulas and using $m^{*} α = β$ gives $⟨ m^{*} x, β^{\lor} ⟩ = ⟨ x, α^{\lor} ⟩$ for every $x \in X (T)$ . The root bijection preserves addition, so the inverse images of $Φ^{+} (B)$ form a positive system in $H$ . Choose its Borel $B_{H}$ by [F7]; it is a product Borel since the central torus has no roots. By [F6] and [F7], $B_{H}$ and $B$ are generated by their maximal tori and positive root groups; their identified generators give $m (B_{H}) = B$ . All highest weights below use these compatible pairs.

4.1givenstep 1.1step 3.1

For every positive root $β = m^{*} α$ of $H$ , step 3.1 gives $⟨ λ_{H}, β^{\lor} ⟩ = ⟨ λ, α^{\lor} ⟩ \geq 0$ . Hence $λ_{H}$ is dominant for the product Borel $B_{H}$ , so the hypothesis of [F2] is satisfied.

5.1F2F4step 4.1

By [F2] there is a rational representation $W$ of $H$ containing a primitive vector $v$ of weight $λ_{H}$ . Let $V_{H} \subseteq W$ be the $H$ -submodule generated by $v$ ; applying [F4] to $V_{H}$ and $v$ , the quotient $Q = V_{H} / V_{H}^{'}$ by the largest proper $H$ -submodule $V_{H}^{'}$ is a simple $H$ -module, and the image of $v$ is a nonzero primitive vector of weight $λ_{H}$ in $Q$ , so $λ_{H}$ is the highest weight of the simple module $Q$ .

6.1F3F5step 1.1step 5.1

By [F3] applied to the central isogeny $m : H \to G$ (with $π = m$ ) and the simple $H$ -module $Q$ of highest weight $λ_{H} \in X (T)$ , the kernel $N$ acts trivially on $Q$ and $Q$ descends along $H / N = G$ to a simple $G$ -module of highest weight $λ_{H}$ , which is $λ$ under the identification $X (T) \subseteq X (T_{H})$ . This descended module is finite-dimensional by [F5], since $G$ is affine of finite type. Hence every dominant $λ$ is the highest weight of a simple finite-dimensional rational representation of $G$ .

7.1F4F5step 6.1∎

For the equivalence: a simple finite-dimensional $G$ -module of highest weight $λ$ contains a primitive vector of weight $λ$ by [F5], while conversely a rational representation containing a primitive vector $v$ of weight $λ$ has, by [F4] applied to the submodule generated by $v$ , a simple quotient of highest weight $λ$ , finite-dimensional by [F5]. Thus the two formulations are equivalent.

Remarks

This is the reduction in Milne's proof of Theorem 22.20: the product $Z (G)_{t} \times G_{d e r}$ is handled by the semisimple case plus the torus factor, and the central isogeny of (19.25) performs the descent.
The hypothesis that $λ$ is a character of $T$ is used exactly through the identification $X (T) \subseteq X (T_{H})$ ; a dominant element of $X (T_{H})$ not lying in $X (T)$ would produce a simple module of the covering group that does not descend.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Dominant weights classify the simple rational representations of a split reductive group

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(G, T)$ be a split reductive group over a field $k$ , let $B \supseteq T$ be a Borel subgroup, and let $X (T)^{+}$ be the set of dominant characters of the maximal torus $T$ (Weights, dominant weights and the highest-weight order of a rational representation). For every $λ \in X (T)^{+}$ there exists a simple rational representation $V (λ)$ of $G$ , unique up to isomorphism, whose $T$ -weight decomposition is $V (λ) = V (λ)_{λ} \oplus ⨁_{μ < λ} V (λ)_{μ}$ with $\dim_{k} V (λ)_{λ} = 1$ , and every simple rational representation of $G$ is isomorphic to $V (λ)$ for a unique $λ \in X (T)^{+}$ (Simple rational representations have a unique highest weight, Simple modules with equal highest weight are isomorphic, Every dominant character of a split reductive group is a highest weight). The map sending a simple representation to its highest weight is a bijection from the set of isomorphism classes of simple rational representations of $G$ to $X (T)^{+}$ ; it holds in every characteristic.

Facts & Assumptions

Given: AC; a split reductive group $(G, T)$ over $k$ with Borel $B \supseteq T$ , and a dominant $λ \in X (T)^{+}$ .

[F1]

Existence of primitive vectors of dominant weight. Every dominant $λ \in X (T)^{+}$ is the highest weight of a simple finite-dimensional rational representation of $G$ , and equivalently there is a rational representation of $G$ containing a primitive vector of weight $λ$ (Every dominant character of a split reductive group is a highest weight).

[F2]

Modules generated by a primitive vector. If a rational representation $W$ of $(G, T)$ is generated as a $G$ -module by a primitive vector $v$ of weight $λ$ , then $W = k v \oplus ⨁_{μ < λ} W_{μ}$ with $W_{λ} = k v$ one-dimensional, and $W$ has a largest proper $G$ -submodule $W^{'}$ , the quotient $W / W^{'}$ being a simple $G$ -module generated by the image of $v$ (Modules generated by a primitive vector, Primitive vectors for a Borel pair).

[F3]

Highest weight of a simple module. Every simple rational representation $V$ of $G$ contains a primitive vector $v$ , unique up to a nonzero scalar, its weight $λ$ is dominant, $V_{λ} = k v$ , every weight $μ$ of $V$ satisfies $μ \leq λ$ , and any two primitive vectors of $V$ have the same weight (Simple rational representations have a unique highest weight).

[F4]

Uniqueness. Simple rational representations of $(G, T)$ with equal highest weight are isomorphic (Simple modules with equal highest weight are isomorphic).

[F5]

Finite dimensionality. Simple rational representations of the affine group scheme $G$ of finite type are finite-dimensional (Simple rational representations are finite-dimensional).

Proof

Given: AC; a split reductive group $(G, T)$ over $k$ with Borel $B \supseteq T$ , and a dominant $λ \in X (T)^{+}$ .

Proof technique: direct.

1.1F1F2F5

Existence: by [F1] there is a rational representation $W$ of $G$ containing a primitive vector $v$ of weight $λ$ . Let $W_{0} \subseteq W$ be the $G$ -submodule generated by $v$ ; by [F2] applied to $W_{0}$ and $v$ , one has $W_{0} = k v \oplus ⨁_{μ < λ} (W_{0})_{μ}$ with $(W_{0})_{λ} = k v$ one-dimensional, and the quotient $V (λ) = W_{0} / W_{0}^{'}$ by the largest proper submodule is a simple $G$ -module generated by the image of $v$ . The image of $v$ is nonzero of weight $λ$ , and the weight spaces of the quotient are the images of the weight spaces of $W_{0}$ , so $V (λ)_{λ}$ is the line generated by the image of $v$ and every other weight $μ$ of $V (λ)$ satisfies $μ < λ$ . By [F5] $V (λ)$ is finite-dimensional.

1.2F3F4

Uniqueness and exhaustiveness: let $V$ be any simple rational representation of $G$ . By [F3] $V$ contains a primitive vector $v$ , unique up to scalar, whose weight $λ_{V}$ is dominant and is the highest weight of $V$ ; the weight $λ_{V}$ is therefore uniquely determined by $V$ . If $V_{1}, V_{2}$ are simple with $λ_{V_{1}} = λ_{V_{2}}$ , then $V_{1} ≅ V_{2}$ by [F4]. Hence the map $V \mapsto λ_{V}$ induces a bijection from the set of isomorphism classes of simple rational representations of $G$ to the set of dominant characters realized as highest weights.

2.1step 1.1step 1.2∎

Combining steps 1.1 and 1.2: for every $λ \in X (T)^{+}$ the module $V (λ)$ of step 1.1 is simple with $V (λ) = V (λ)_{λ} \oplus ⨁_{μ < λ} V (λ)_{μ}$ and $\dim_{k} V (λ)_{λ} = 1$ , and any simple rational representation is isomorphic to $V (λ)$ for the unique $λ \in X (T)^{+}$ given by its highest weight. No step used a hypothesis on the characteristic of $k$ , so the classification holds in every characteristic.

Remarks

The two halves of the argument are independent: existence comes from the construction of a primitive vector of weight $λ$ followed by the quotient by the largest proper submodule, and uniqueness comes from the comparison of two simple modules with the same highest weight.
No separability, characteristic-zero, or algebraic-closure hypothesis appears, in accordance with Milne's Theorem 22.2; the only finiteness input is that simple rational representations of a finite-type affine group scheme are finite-dimensional, used to make $V (λ)$ finite-dimensional.

RemarkRemark: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

The highest-weight classification does not imply semisimplicity in positive characteristic

Remarks

Assume the Axiom of Choice inherited from the named suppliers (The Axiom of Choice). The classification of Dominant weights classify the simple rational representations of a split reductive group holds for a split reductive group over every field, but it does not imply that every rational representation is semisimple. Complete reducibility is a characteristic-zero phenomenon (Complete reducibility of rational modules in characteristic zero); in characteristic $p > 0$ a split reductive group need not be linearly reductive, as the companion counterexample Rational modules need not be semisimple in characteristic p ↗ on this pair's examples page shows (Milne Example 12.55 and Exercise 12-9; Steinberg Ch. 12, the paragraph after Theorem 39(e)). Readers should not infer semisimplicity of $Rep (G)$ from the existence and uniqueness of simple modules with prescribed dominant highest weight.

The two statements concern different properties: the classification theorem only asserts that simple modules are parametrized by dominant weights and that the top weight space is one-dimensional, while semisimplicity of every finite-dimensional rational representation is a strictly stronger property that fails already for ${S L}_{2}$ in characteristic $p > 0$ (Split reductive groups).

5 · Examples, counterexamples and false statements

None yet.