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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.

Depends on

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Robert Steinberg, Lectures on Chevalley Groups (Yale University, 1967; notes prepared by J. Faulkner and R. Wilson) (standard reference, not scraped)