## Galois action on one-dimensional quotients of l-adic cohomology

Let $A$ be an abelian variety of dimension $g$ over a number field $K$, and $\ell$ be a rational prime. Suppose that the Galois action on the $\ell$-adic cohomology $H^k(A, \mathbb{Z}_\ell) \otimes_{\mathbb{Z}_\ell} \overline{\mathbb{Q}_\ell}$ has a one-dimensional Jordan-Holder quotient. Is there a (conjectural or known) classification of the possible characters $\psi : Gal(K^{ab} / K) \to \overline{\mathbb{Q}_\ell}^*$ which can give the action on this quotient (in terms of $K$, $g$, and $\ell$)?

(When $g = 1$, there is a simple explicit description: For $H^0$ and $H^2$, it must always be the trivial and cyclotomic character respectively. For $H^1$, there can only be a one-dimensional Jordan-Holder quotient if the elliptic curve has CM, in which case there is a well-known explicit description. This follows from Serre's open image theorem, since any reducible subgroup of $GL_2(\mathbb{Z}_\ell)$ has infinite index.)

Even if there is not a good description in full generality, are there any interesting classes of examples (besides those examples coming from abelian varieties of CM-type) which admit an explicit description?

EDIT: I would like to know not just whether a particular $\psi$ can occur for some abelian variety over some number field (which is answered below by Aşağı Güzdək), but rather for which pairs $(K, g)$ it can occur for an abelian variety of dimension $g$ defined over the number field $K$.

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