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Consider an abelian variety $A$ over a number field, and look at the representation of its Mumford-Tate group on $H^1(A)$, restricted to the commutator subgroup. Is it possible that every element of this commutator subgroup preserves some element of $H^1(A)$, without this representation containing a copy of the trivial representation (i.e. for it to look like the standard representation of $SO_3(\mathbb{R})$ on $\mathbb{R}^3$)?

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Possible commutator subgroups of Mumford-Tate groups were classified by Satake, Satake, Ichirô, Symplectic representations of algebraic groups satisfying a certain analyticity condition. Acta Math. 117 1967 215–279, see also Deligne, Pierre, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 247–289, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979. –  Mikhail Borovoi Apr 20 '11 at 22:43
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