Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which exists in a world where standard conjectures and anything else you can expect is true, in the spirit of Grothendieck, and like in Serre's paper in Motives, PSPM 55, volume 1 – a paper which unfortunately does not seem to be online except partially on Amazon).
Recall that the motivic galois group of $E$, denoted $G_{M(E)}$, is defined as the tensor-automorphic group
of the functor "Betti realization" from the category $M(E)$, defined as the smallest Tannakian sub-category of the category of motives containing $E$, to the category of $\mathbb Q$-vector spaces.
An equivalent (under the conjectures of Tate and Hodge)
and more concrete (at least for me) definition is as follows (see the same paper of Serre):
let $H_b(E)$ be the Betti realization of $E$ (a finite-dimensional $\mathbb Q$-vector space),
and let $\rho_\ell$ be the $\ell$-adic realization of $E$, which is a continuous semi-simple
representation $\rho_\ell$ of $\Gamma_{\mathbb Q}$ (the absolute Galois group of $\mathbb Q$) over $H_b(E) \otimes \mathbb Q_\ell$. Then $G_{M(E)}$ is the algebraic subgroup $Gl(H_b(E))$ such that $G_{M(E)} \otimes \mathbb Q_\ell$ is the Zariski-closure of the image of $\rho_\ell$, for every prime number $\ell$.
In any case, $G_{M(E)}$ is a reductive group. My question is
What reductive groups over $\mathbb Q$ arise as $G_{M(E)}$ for some pure motive $E$?
This question is asked by Serre in the section 8 of his paper. At that time there was not much known on it apparently (it was 20 years ago): Serre notices that such a group $G$ must have an element $\gamma \in G(\mathbb R)$ such that $\gamma^2=1$ and whose centralizer is a maximal compact subgroup of $G(\mathbb R)$. This necessary conditions implies that $Sl_2$ is not
a motivic Galois group. On the positive side, I believe that symplectic group $GSp_{2n}$
are known to be motivic Galois group (attached to $E=$ an abelian variety over $\mathbb Q$ of dimension $n$, sufficiently generic), and I know a few other examples like those attached to
elliptic curves with complex multiplication.
There has been a lot of work and progresses on motives since then, and I am asking to know
what progresses have been made in the direction of this question. I would be very surprised
if it had been solved completely (even assuming the standard conjectures + Hodge conjecture + Tate conjecture), but I would be interested in any partial results:
For example, what if we just try to classify the connected reductive group which arise as neutral component
of a $G_{M(E)}$? those groups up to isogeny ? those groups after extension to the algebraic closure of $\mathbb Q$? Conjectural answers (conjectural in the sense that they are not proved even assuming the conjectures assumed above) are also more than welcome.
Even a list of example of groups that are provably or possibly motivic Galois group which goes beyond the small list I have given would be very useful. And of course, any references more recent than Serre's could be very useful.
Best Answer
The first question (applied to $\mathrm{GL}(2)$-abelian varieties over $\mathbf{Q}$) seems to include the following problem: what totally real fields $F$ occur as the field of coefficients of a classical weight $2$ modular form? This seems a totally impossible question to answer. For example, it includes the question of which Hilbert modular surfaces $X_F$ have rational points; since $X_F$ is of general type for $F$ of suitable large discriminant, the answer seems hard to predict in advance (especially because fields $F$ of arbitrarily large degree do actually occur).
For the second, surely the work of Zhiwei Yun (http://arxiv.org/pdf/1112.2434v1.pdf) is relevant here.