Primitive vectors for a Borel pair

Let $(G,T)$ be a split reductive group, $B\supseteq T$ a Borel subgroup with unipotent radical $U=B_u$ and roots $\mathrm{\Phi }$ (Borel subgroups, maximal tori and Borel pairs, Unipotent algebraic groups and unipotent representations, Roots and root groups of a split reductive group, Root subgroups of a split reductive group). Let $(V,r)$ be a rational representation (Rational representations and comodules of an affine group scheme). A nonzero vector $v\in V$ is primitive (for the pair $(B,T)$) if it spans a $B$-stable line. Such a vector is fixed by $U$ and is a $T$-eigenvector: the action on its one-dimensional line restricts to a character of $T$, and unipotence forces the action of $U$ on that line to be trivial. Assuming the Axiom of Choice (The Axiom of Choice) for the cited split-Borel structure, multiplication is an isomorphism $U⋊T\to B$, as justified in the Remarks below. Consequently the converse holds as well: a nonzero $U$-fixed $T$-eigenvector is primitive. The weight of a primitive vector is the character $\lambda \in X(T)$ with $t\cdot v=\lambda (t)v$ for all $t$ and all $k$-algebras; for a finite-dimensional $V$ it is the weight $\lambda$ with $v\in V_{\lambda }$ (Weights, dominant weights and the highest-weight order of a rational representation).

• Weight and converse. Restriction to a fixed line gives a unique character of $T$, so its weight is well defined without a choice principle. The forward implication uses only the defining fixed-vector property of a unipotent group applied to that line. For the converse, under AC, the root-group coordinates give $\dim U=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$, while the adjoint weight decomposition gives $\dim B=\dim T+\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$. The intersection $U\cap T$ is trivial by A subgroup that is both unipotent and diagonalizable is trivial. Since $U$ is normal in $B$, multiplication gives a homomorphism $U⋊T\to B$ with trivial scheme kernel; the exact-image theorem Group images are exact kernel quotients and preserve affine smooth connected properties identifies its source with a smooth connected closed image of the same dimension as the smooth connected group $B$. Thus that image is $B$, proving $B=U⋊T$. A $U$-fixed $T$-eigenline is consequently $B$-stable.
• Normalisation. The Borel subgroup is the one used to define the positive system ${\mathrm{\Phi }}^{+}$, so a primitive vector is fixed by the unipotent group of positive root subgroups. The opposite unipotent group $U^{-}$ generates the big cell with $B$ and is used only in the proofs of the weight statements.
• Choice scope. The definition by a $B$-stable line, its weight, and the forward implication above are choice-free. AC is inherited only for the supplemental structural converse and the root-group facts used to describe the opposite big cell.