I'm trying to figure out what a very rough sketch of the Langlands program could be. From what I (think I) understand, objects called reductive algebraic groups together with related so-called automorphic representations thereof as morphisms should form a category which in this context should correspond to another category whose objects are called L-groups, so that there should exist a functor from the former category to the latter.
My first question is: is this true? The second one is: has this functor been given some special name (like the Langlands functor or something the like)? And finally, has the automorphism group of this functor, viewed as the fiber functor of a covering of the first category by the second (since Grothendieck, from what I've been told, considered such a group to be the general definition of a fundamental group) been considered so far? What about its (conjectural) structure?
Many thanks in advance.
The (classical) Langlands correspondence is still being pieced together and is not so precise that it can be thought of as a specific functor between well-defined categories.