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Let $(V,\Phi)$ be a root system with dual root system $(V^{\ast},\Phi^{\vee})$. Let $\Delta = \{\alpha_1, ... , \alpha_n\}$ be a set of simple roots for $V$, and let $\Delta^{\vee} = \{\alpha_1^{\vee}, ... , \alpha_n^{\vee}\}$ be the coroots corresponding to $\Delta$.

We have the fundamental weights $\hat{\Delta} = \{ \omega_1, ... , \omega_n\}$, which is by definition the dual basis of $\Delta^{\vee}$, and we have the fundamental coweights $\hat{\Delta}^{\vee} = \{ \omega_1^{\vee}, ... , \omega_n^{\vee}\}$, which is by definition the dual basis of $\Delta$.

When $\Phi$ is reduced, the lattices $$\{v \in V : \langle v, \alpha^{\vee} \rangle \in \mathbb Z \textrm{ for all } \alpha^{\vee} \in \Phi^{\vee} \}$$ and $$\mathbb Z \omega_1 \oplus \cdots \mathbb Z \omega_n$$ are equal, and this is called the weight lattice. But for $\Phi$ not necessarily reduced, the second lattice is in general a proper subset of the first. This is because although $\Delta$ is a base of $\Phi$, $\Delta^{\vee}$ is not in general a base of $\Phi^{\vee}$ (since if $\alpha \in \Delta$ is such that $2\alpha$ is a root, then $\alpha^{\vee}$ is divisible by $(2\alpha)^{\vee}$).

How should the weight lattice be defined in the nonreduced case? Should we even bother calling the $\omega_i$ "fundamental weights?" I know that $\hat{\Delta}$ and $\hat{\Delta}^{\vee}$ are still dealt with in the nonreduced case, for example in the proof of Langlands classification here (section 2.2).

A possible alternative around this is to define $\Delta^{\vee}$ not as the set of coroots $\alpha^{\vee}$ for $\alpha \in \Delta$, but rather as the set of simple roots of $\Phi^{\vee}$ corresponding to same chamber as $\Delta$.

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Bourbaki has the most detailed treatment, but they tend not to deal with weight lattices (or co-weight lattices) so explicitly outside their account of some of the representation theory. Thus you can make any definition you like.

The basic question here is what your motivation is. Weights arise in representation theory, whereas non-reduced root systems come up in working (as Borel-Tits did) over non-algebraically closed fields as fields of definition for reductive groups.

[A small typographic point is that the symbol "omega" is often used in place of the symbol written in LaTeX as \varpi (and used frequently by Bourbaki): a variation of the handwritten "pi" for "poids". While "omega" is a harmless choice, it fails to reinforce the history of the notation]

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One can define the weight lattice of any root systems, reduced or not, as the coweights of the dual root system, see section 7, in particular Definition 7.1, of my AMS memoirs with Ottmar Loos. The example of a non-reduced root system is worked out in 8.7 of the memoirs. It can be downloaded here

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