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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.

Depends on

Used by

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

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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)