Weights, dominant weights and the highest-weight order of a rational representation

Let $k$ be a field, let $(G,T)$ be a split reductive group over $k$ with Weyl group $W$ and root datum $(X,\mathrm{\Phi },\alpha \mapsto {\alpha }^{\vee })$, $X=X(T)$, and let $B\supseteq T$ be a Borel subgroup with system of positive roots ${\mathrm{\Phi }}^{+}$ and base $\mathrm{\Delta }$ (Split reductive groups, Roots and root groups of a split reductive group, Root subgroups of a split reductive group, Character and cocharacter lattices of a split torus). For a rational representation $(V,r)$ of $G$ (Rational representations and comodules of an affine group scheme) the $T$-weight decomposition is $V={⨁}_{\chi \in X}{V}_{\chi }$ with $V_{\chi }=\{v\in V:\rho (v)=v\otimes e_{\chi }\}$ (Representations of diagonalizable groups split into character eigenspaces); the weights of $(V,r)$ are the characters $\chi$ with $V_{\chi }\mathrm{\ne }0$, and $V_{\chi }$ is the weight space of $\chi$. An element $\lambda \in X$ is dominant if $⟨\lambda ,{\alpha }^{\vee }⟩\ge 0$ for every $\alpha \in {\mathrm{\Phi }}^{+}$ (equivalently, for every $\alpha \in \mathrm{\Delta }$); write $X^{+}$ for the set of dominant weights. Define a partial order on $X$ by $\lambda \ge \mu$ if and only if $\lambda -\mu ={\sum }_{\alpha \in \mathrm{\Delta }}{m}_{\alpha }\alpha$ with $m_{\alpha }\in {\mathbb{Z}}_{\ge 0}$. The fundamental weights ${\omega }_i$ ($i\in \mathrm{\Delta }$) are the unique elements of $\mathbb{Q}\mathrm{\Phi }\subseteq X{\otimes }_{\mathbb{Z}}\mathbb{Q}$ dual to the simple coroots, $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$; they form a basis of $\mathbb{Q}\mathrm{\Phi }$ and every $\lambda \in X\otimes \mathbb{Q}$ decomposes as $\lambda ={\sum }_i⟨\lambda ,{\alpha }_i^{\vee }⟩{\omega }_i+{\lambda }_0$ with ${\lambda }_0$ pairing to $0$ with every coroot (Combinatorics of a reduced root datum).

• Conventions. Dominance is tested on the positive system ${\mathrm{\Phi }}^{+}$, which is the system of roots of $B$; the equivalence of testing on ${\mathrm{\Phi }}^{+}$ or on the base $\mathrm{\Delta }$ holds for coroot pairings as follows. Average an inner product on $X\otimes \mathbb{R}$ over the finite Weyl group of Combinatorics of a reduced root datum. Each root reflection is then orthogonal, so this inner product identifies ${\alpha }^{\vee }$ with $2\alpha \mathrm{/}(\alpha ,\alpha )$. If $\alpha ={\sum }_i{n}_i{\alpha }_i$ is positive, then ${\alpha }^{\vee }={\sum }_i{n}_i({\alpha }_i,{\alpha }_i)\mathrm{/}(\alpha ,\alpha ){\alpha }_i^{\vee }$, a nonnegative combination of simple coroots. Thus testing the simple coroots suffices. The order is generated by the positive simple roots, so $\lambda \ge \mu$ is the relation used for highest weights.
• Existence and uniqueness. In that inner product the simple-coroot functionals are independent on the root span, since the simple roots form a basis. Their pairing matrix is rational and invertible, so their dual vectors ${\omega }_i$ lie uniquely in $\mathbb{Q}\mathrm{\Phi }$. Subtracting ${\sum }_i⟨\lambda ,{\alpha }_i^{\vee }⟩{\omega }_i$ from $\lambda$ gives the displayed ${\lambda }_0$, which annihilates all coroots by the preceding coroot expansion. Specifying the root span is essential when the central torus is nontrivial.
• Fundamental weights need not lie in $X$. The ${\omega }_i$ are elements of the rational vector space $X\otimes \mathbb{Q}$; the correction term ${\lambda }_0$ in the displayed decomposition lies in the annihilator of the coroot lattice, and it need not belong to $X$, nor need ${\sum }_i⟨\lambda ,{\alpha }_i^{\vee }⟩{\omega }_i$ lie in $X$ for an arbitrary reductive root datum. This is why the classification below is proved through the product-with-a-torus and central-isogeny route rather than by adding a lattice correction term directly.
• No Choice. The definition only records eigenspace decompositions and lattice dualities of the cited suppliers; no choice principle is used.