Langlands correspondence for higher local fields? Let $F$ be a one-dimensional local field. Then Langlands conjectures for $GL_n(F)$ say (among other things) that there is a unique bijection between the set of equivalence classes of irreducible admissible representations of $GL_n(F)$ and the set of equivalence classes of continuous Frobenius semisimple complex $n$-dimensional Weil-Deligne representations of the Weil group of $F$ that preserves $L$-functions and $\epsilon$-factors. This statement has been proven for all one-dimensional local fields. 
My question is: is it possible to formulate any meaningful analogue of Langlands conjectures for higher local fields (e.g. formal Laurent series over $\mathbb{Q}_p$)? If this is possible, conjectures for $GL_1$ should probably be equivalent to higher local class field theory (it says that for an $n$-dimensional local field $F$, there is a functorial map
$$
K_n(F)\rightarrow \mathrm{Gal}(F^{ab}/F)
$$
from $n$-th Milnor K-group to the Galois group of maximal abelian extension, which induces an isomorphism $K_n(F)/N_{L/F}(K_n(L))\rightarrow \mathrm{Gal}(L/F)$ for a finite abelian extension $L/F$).
Frankly, I do not even know what should be the right definition of Weil group of a higher local field (nor did my literature search give any results) but maybe other people have figured it out.
 A: Similar and related questions are: 
On Geometric Langlands Correspondence
Langlands conjectures in higher dimensions
Kapranov's analogies
probably more...

I've already wrote on subj here: https://mathoverflow.net/q/131884
Let me just add a few words.
1) Abelian case of higher dimensional Langlands (=class field theory) developped by A.N. Parshin (1975) and K.Kato (1977) and later on by Fesenko and others (survey 2000). 
2) Around 1992-5 Mikhail Kapranov wrote quite a speculative paper "Analogies between the Langlands correspondence and topological quantum field theory"
(See: Kapranov's analogies )
One of his ideas is the following:
Instead correspondence between Rep(Gal) <-> Rep( G(Adelic) ),
one should consider "higher represenations" (representations not into category of vector spaces, but to k-categories). 
So Kapranov's idea:  n-th dimensional k-representations of dimension r of Galois group should correspond to (n-k)-representations of GL_r(n-Local Field)
That will include Parshin-Kato abelian case as a subcase as Kapranov explained.
However actual work with higher representations is somewhat elusive, 
so difficult to transform insights into precise theorems/conjectures. 
A: The Langlands correspondence for higher local fields is still at an early stage of development.  I haven't really kept up with it, but here's some key points.
As the question stated, and Loren commented, the starting point is the $GL_1$ case, which is class field theory for higher local fields.  Local class field theory relates the abelianized Galois group $Gal_F^{ab}$ of a local field $F$ to the multiplicative group $F^\times = K_1(F)$.  For a higher local fields $E$, Kato's class field theory relates the abelianized Galois group $Gal_E^{ab}$ to the Milnor K-group $K_n(E)$.  
For example, let $E = {\mathbb Q}_p((t))$.  Then there's a canonical homomorphism $\Phi \colon K_2(E) \rightarrow Gal_E^{ab}$ such that for all finite abelian $L/E$, $\Phi$ induces an isomorphism from $K_2(E) / N_{L/E} K_2(L)$ to $Gal(L/E)$.  This gives a bijection between finite abelian extensions of $E$ (in a fixed algebraic closure) and open, finite-index subgroups of $K_2(E)$.  This is the main theorem described in 
Kato, Kazuya, A generalization of local class field theory by using K-groups. I, Proc. Japan Acad., Ser. A 53, 140-143 (1977). ZBL0436.12011.  
You can look at this paper to see the topology on $K_2(E)$ and more details.  In particular, this suggests a possible Weil group for $E$.  Namely, Kato reciprocity gives an isomorphism from a completion of $K_2(E)$ to $Gal_E^{ab}$.  One might let the abelianized Weil group be the subgroup $Weil_E^{ab}$ of $Gal_E^{ab}$ corresponding to the uncompleted $K_2(E)$.  And perhaps the (nonabelian) Weil group should be defined by pulling back.  I.e., look at the map $\pi \colon Gal_E \rightarrow Gal_E^{ab}$, and define $Weil_E = \pi^{-1}(Weil_E^{ab})$. I haven't explored if this is the right idea though.  
Kato goes beyond this, from 2-dimensional to n-dimensional local fields, and from $K_2$ to $K_n$ accordingly.  These aren't hard to find, and there are surveys floating around.  See the Invitation to Higher Local Fields volume, for example.  Even $K_2$ is interesting, I think!
Note that Kato's paper was from 1977... so what about the Langlands program for fields like $E$?  A natural first step is figuring out a suitable version of the Satake isomorphism, and the Iwahori-Hecke algebra.  There's a series of papers by Kazhdan, Gaitsgory, Braverman, Patnaik, Rousseau, Gaussent (and certainly others) on the subject.
Recent landmark papers are 


*

*Braverman, Alexander; Kazhdan, David, The spherical Hecke algebra for affine Kac-Moody groups. I, Ann. Math. (2) 174, No. 3, 1603-1642 (2011). ZBL1235.22027. 

*Gaussent, Stéphane; Rousseau, Guy, Spherical Hecke algebras for Kac-Moody groups over local fields., Ann. Math. (2) 180, No. 3, 1051-1087 (2014). ZBL1315.20046. 

*Braverman, Alexander; Kazhdan, David; Patnaik, Manish M., Iwahori-Hecke algebras for $p$-adic loop groups, Invent. Math. 204, No. 2, 347-442 (2016). ZBL1345.22011..  


Note that a group like $SL_2(E)$ can be seen as a loop group over ${\mathbb Q}_p$.  Hence the appearance of words like "loop group" and "Kac-Moody group". 
The Langlands dual group certainly arises in these studies, but I haven't seen something quite as straightforward as a parameters from the Weil group (described above) to the dual group.  I haven't looked too hard either, so maybe it's in there somewhere.  There seems to be a fancier, more categorical, parameterization involved.  I'd be tempted to bring it down to earth a bit, following Kato.
The other direction that I haven't seen -- and one that I think is worth pursuing -- is the case of (nonsplit) tori.  That's important for any putative Langlands program, and should require an interesting mix of Milnor K-theory and Galois cohomology.
