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Let $G$ be a split reductive group over a field $K$ with a maximal torus $T$. The extended affine Weyl group is defined to be the semidirect product of the finite Weyl group $W$ and the cocharacter lattice $X_{*}(T)$. There is another definition of extended affine Weyl groups where we take the semidirect product of $W$ with the coweight lattice. Are these two definitions equivalent with some conditions on $G$ and if not, which definition does one use in practice?

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This is the same definition, I believe. Coweights are by definition cocharacters of the maximal torus.

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  • $\begingroup$ By coweights, I meant the abstract coweights, which are defined in an analogous manner as abstract weights. $\endgroup$
    – Math0097
    Commented Nov 14, 2023 at 22:38

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