Relationship between motivic Galois groups and Langlands program I would like to know if there is any relationship between the motivic Galois groups and the Langlands program.
Many thanks.
 A: Take $K=\mathbb{Q}$ for simplicity, but this applies to any number field $K$, and let $L$ be the Langlands group and $\mathcal{G}$ the motivic Galois group of $\mathbb{Q}$.
Then the conjectural relation between automorphic forms and motives (the Langlands program) implies that there is an homomorphism (up to a certain conjugation) such that the diagram
$$\require{AMScd}\begin{CD}
L @>{}>> \mathcal{G};\\
@VVV @VVV \\
\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) @>{}>>\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q});
\end{CD}$$
is commutative.
To get a better idea of both sides of this picture it is useful to use some classic (and better understood) intermediate groups. Let $W$ be the Weil group, $\mathcal{S}$ the Serre group and $\mathcal{T}$ the Taniyama group, then we have something like this
$$L \longrightarrow W \longrightarrow \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(automorphic side)}$$
$$\mathcal{G} \longrightarrow \mathcal{S} \longrightarrow \mathcal{T} \longrightarrow \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(motivic side)}$$
Most of this goes back to


*

*Robert Langlands, "Automorphic representations, Shimura varieties and motives" (1977)


Some other relevant references are


*

*James Milne, Kuang-yen Shih, "Langlands's Construction of the Taniyama Group" (1982)

*Norbert Schappacher, "CM motives and the Taniyama group" (1994)

*James Arthur, "A note on the automorphic Langlands group" (2002)
