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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 $Z (G)_{t} \times G_{d e r}$ .

Depends on

Used by

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