I would like to know if there is any relationship between the **motivic Galois groups** and the **Langlands program**.


Many thanks.



Relationship between motivic Galois groups and Langlands program
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](https://publications.ias.edu/sites/default/files/autoreps-ps.pdf)" (1977)


Some other relevant references are


* James Milne, Kuang-yen Shih, "[Langlands's Construction of the Taniyama Group](http://www.jmilne.org/math/articles/1982c.pdf)" (1982)
* Norbert Schappacher, "[CM motives and the Taniyama group](http://www-irma.u-strasbg.fr/~schappa/NSch/Publications_files/1994a_TaniyamaGrp.pdf)" (1994)
* James Arthur, "[A note on the automorphic Langlands group](https://www.math.ubc.ca/~cass/arthur/pdf/automorphic-langlands-group.pdf)" (2002)


