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Let $G$ be a semisimple algebraic group defined over $\mathbb{Q}$ and $(\rho,V)$ be an irreducible $\mathbb{Q}$-representation of $G$ preserving a sympectic form $w$ defined over $\mathbb{Q}$.

Can one find a $w$-isotropic subspace $L$ of $V$, defined over $\mathbb{Q}$ and stabilized by a (one dimensional) real split torus of $G$ ?

I would appreciate any reference to similar questions.

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    $\begingroup$ No. Let $K=\mathbb Q(i)$, $G=SU_{2,K/ \mathbb Q}$, $\rho$ the natural 4-dimensional representation of $G$ in $R_{K/ \mathbb Q} K^2$. Then $G_{\mathbb R}$ is anisotropic, hence has no nontrivial split tori. Hence there is no $w$-isotropic subspace stabilized by a real split torus of $G$. $\endgroup$
    – Mikhail Borovoi
    Commented Jan 25, 2017 at 12:07

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