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.



Langlands correspondence for higher local fields?
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**](http://dx.doi.org/10.3792/pjaa.53.140), Proc. Japan Acad., Ser. A 53, 140-143 (1977). [ZBL0436.12011](https://zbmath.org/?q=an:0436.12011). 


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 


1. *Braverman, Alexander; Kazhdan, David*, [**The spherical Hecke algebra for affine Kac-Moody groups. I**](http://dx.doi.org/10.4007/annals.2011.174.3.5), Ann. Math. (2) 174, No. 3, 1603-1642 (2011). [ZBL1235.22027](https://zbmath.org/?q=an:1235.22027).
2. *Gaussent, Stéphane; Rousseau, Guy*, [**Spherical Hecke algebras for Kac-Moody groups over local fields.**](http://dx.doi.org/10.4007/annals.2014.180.3.5), Ann. Math. (2) 180, No. 3, 1051-1087 (2014). [ZBL1315.20046](https://zbmath.org/?q=an:1315.20046).
3. *Braverman, Alexander; Kazhdan, David; Patnaik, Manish M.*, [**Iwahori-Hecke algebras for $p$-adic loop groups**](http://dx.doi.org/10.1007/s00222-015-0612-x), Invent. Math. 204, No. 2, 347-442 (2016). [ZBL1345.22011](https://zbmath.org/?q=an:1345.22011)..


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.


