Note that a global field is dense in any finite product of its local fields. Take, for example, the global field $F_q(T)$ of $P_{F_q}^1$ and its local fields at $T = 0$ and $T = \infty$.

I agree with @dhy about the relationship between Frenkel-Gaitsgory and archimedean Langlands. I think I misunderstood the question a bit. Anyways, I think I should look at the paper of Ben-Zvi and Nadler!

Over C, see work of Edward Frenkel and Dennis Gaitsgory. E.g., Frenkel's book "Langlands Correspondence for Loop Groups". Over R, I recall seeing something by Nadler or Ben-Zvi.

See en.wikipedia.org/wiki/Littelmann_path_model#Branching_rule. You're branching to a Levi subgroup.

Thanks -- I knew about Parshin's later work on higher adeles, but not about this 1975 work. Does he give full proofs for higher local class field theory? I don't read Russian, but I can't see complete proofs in the 1975 paper you linked.

Consider $G = SL_2(F_2)$ and $H = SL_2$ and $q=2$ and $\rho$ the identity.

In this spirit, you might be interested in the monodromic proof of the impossibility of trisection, written by Terry Tao here: terrytao.wordpress.com/2011/08/10/…

What are the sets $I \subset [n]$ in these cases? I'd want to know if there's any pattern among the parabolics.

I’m curious to see if there’s a stronger result too!

Ahh - I messed up on that one. I've made a correction, and sadly it makes the answer a bit weaker when we don't have such a nice bound on $t$. It was too good to be true.