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](http://www2.math.kyushu-u.ac.jp/~takuya/papers/Langlclassif.pdf) (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 $\Delta$.



Definition of the weight lattice for a nonreduced root system
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] 


