As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept!

In any case, I wish now to relate it to yet another topic of which I have only a tenuous grasp: the Langlands program. As I understood more and more about Tannakian formalism, it seemed more and more like it has something to do with Langlands. A google search confirms this. There are several sources that group these two things together. Here is a sample:

http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf

http://www.institut.math.jussieu.fr/~harris/Takagi.pdf

But my understanding of Langlands is weak. I'm certainly familiar with Class Field Theory, and to some limited extent with Taniyama-Shimura. I always found Langlands difficult to penetrate. But now that I know that there is a relationship between Langlands and Tannakian formalism, I am hopeful that this will give me a bird's eye view of Langlands.

So the question is: Does Tannakian formalism simplify the statement of Langlands, or at least motivate it? Does it have to do with the motivic Galois group (defined to be the group predicted from Tannakian formalism on the category of numerical motives)? How precisely is Tannakian formalism used in Langlands?

In light of these ideas, I ask a secondary question: is there a relationship between the standard conjectures and Langlands? (does one imply the other?)

Ididn't understand all the answers, probably there were other readers who did. – Tom Leinster Aug 3 '11 at 1:04