Though there are several automorphic papers discussing the Tannakian outlook (notably Ramakrishnan's article in Motives (Seattle 1991, AMS) and Arthur's [A note on the Langlands group,][1] (referred to above) there is as yet no formulation of Langlands correspondence between Galois representations and automorphic representations as an equivalence of Tannakian categories. There are (at least) two outstanding fundamental questions on the Tannakian aspects of the Langlands correspondence.
- What is the definition of the category of automorphic representations for any number field?
here one means automorphic representations for GL, any $n \ge 0$.
- How to endow the category in 1) with a tensor structure so as to render it Tannakian?
But the Tannakian outlook, despite its present inaccessibility, has already made a profound impact. See Langlands paper ["Ein Marchen etc"][2] (where the Tannakian aspect was first written out with many consequences for the Taniyama group ([Milne's notes)][3]) as well as [Serre's book Abelian l-adic representations][4] for many references.
This is just a rough answer from a novice..for a precise and detailed answer, let us wait for the experts! [1]: http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf [2]: http://publications.ias.edu/rpl/section/26 [3]: http://jmilne.org/math/CourseNotes/cm.html [4]: http://books.google.com/books?id=CLlkdN2lh8cC