I guess it's subjective which part is the meat and which the potatoes! I agree there are deep geometric reasons for all these things over $\mathbb{F}_q(t)$. And lots of people hope this will help in the number field case. But I think that what makes number theory really number theory (and not "just" geometry") is this excess complexity. And phrasing the excess as "carrying" might be helpful and not just cute.

@IlyaBogdanov: Thank you for getting straight to the point :) Yes, I think you're right. And I've updated the question significantly to give a precise conjecture. Your comment suggests something deeper than a bit of real analysis is needed, if my conjecture is indeed true.

@GTA -- Savin describes an isomorphism between the Iwahori-Hecke algebra of a covering group (e.g. an d-fold cover of SL_n) and the Iwahori-Hecke algebra of a related linear group (e.g., a linear quotient of SL_n, determined by d and n). It turns out that this isomorphism reflects an isomorphism of L-groups (using my L-group for the covering group), so it's an instance of functoriality. Presumably some of these may arise from global correspondences between automorphic forms on a covering and a linear group.

@GTA -- Well, sort of. First, one has to have an L-group for covering groups, so that functoriality makes sense. And then one can check that various correspondences are functorial. Examples include the metaplectic correspondence (e.g., recent work of Gan and Savin), and Hecke algebra correspondences. For theta correspondences, there are interesting cases to check functoriality -- e.g., $SL_3$ and a double cover of $SL_3$ (or quasisplit $SU_3$ and its double-cover), which form a dual pair in a double cover of $F_4$. Wee Teck Gan and I may finish that someday.....

@PeterMcNamara -- that sounds promising, thanks! I'm going to take a little while to digest the Jacobson density theorem, and will write to you if I can't put it together.

Well, I know the first part well. But I use the converse of Schur's Lemma to prove irreducibility of the tensor product. One can prove $End_{G \times H}(\pi \boxtimes \rho$ is ${\mathbb C}$ using Schur's Lemma... but then what? Or am I missing something easy here?

I think this works. So in the context of complex representations, let $G = H = {\mathbb C}(T)^\times$ acting on the complex vector space $V = {\mathbb C}(T)$. It's kind of interesting to me to see if there's a countable-dimension example (over the complex numbers), but I won't move the goalposts here!