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I'm asking for a proof or references of the following claim:

Let $V$ be a rational Hodge structure having CM in the sense that its Mumford-Tate group is abelian. Then there is a filtration $F^{\bullet}_{\overline{\mathbb{Q}}}$ on $V_{\overline{\mathbb{Q}}}$ such that the Hodge filtration $F^{\bullet}$ of $V_{\mathbb{C}}$ is given by $$F^{\bullet}=F^{\bullet}_{\overline{\mathbb{Q}}}\otimes_{\overline{\mathbb{Q}}}\mathbb{C}.$$

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A CM Hodge structure is a polarizable rational Hodge structure whose Mumford-Tate group is commutative, hence a torus $T$. The Hodge filtration is split by a cocharacter of $T$ over $\mathbb{C}$, which is automatically defined over the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$.

Added: it's the cocharacter $\mu$ such that $\mu(z)$ acts on $V^{p,q}$ as $z^{-p}$. A cocharacter over $\mathbb{C}$ of a torus defined over $\mathbb{Q}^{al}$ is automatically defined over $\mathbb{Q}^{al}$. Since the cocharacter determines the filtration, it also is defined over $\mathbb{Q}^{al}$.

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  • $\begingroup$ By "a cocharacter", do you mean the one $\mathbb{G}_m\to M_\phi$ induced by $\phi:\mathbb{S}(\mathbb{R})\to GL(V)(\mathbb{R})$? How to use it? On the other hand, shouldn't a representation of a torus automatically decomposes into characters? Why I need a cocharacter? $\endgroup$
    – Syu Gau
    Commented May 22, 2019 at 4:38

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