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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.

Depends on

Used by

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

Dependency tree · two levels

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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)