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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.

Depends on

Used by

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