# 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.

• The beginning of the non-Abelian Langlands conjectures (which, of course, go far beyond $\mathrm{GL}_n$) is the Abelian Langlands conjecture, i.e., class-field theory (the case $n = 1$; see mathoverflow.net/questions/66500/…). I don't know the CFT situation for higher-dimensional local fields, but that's probably the place to start. – LSpice Jun 16 '18 at 0:59

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

1. 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.
2. 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.
3. 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.

• A.N. Parshin developped ideas on higher dimensional class field theory starting at 1975, prior to Kato. See e.g. A. N. Parshin, Class fields and algebraic K-theory, Uspekhi Mat. Nauk, 1975, Volume 30, Issue 1(181), 253–254 mathnet.ru/links/2832bbd4fa4a8796f3ec44aa0b3657ee/rm4196.pdf and general overview of Parshin's works: researchgate.net/publication/… – Alexander Chervov Jun 16 '18 at 19:55
• 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. – Marty Jun 16 '18 at 21:27
• @Marty that short paper of Parshin has no proofs in it. The paper is a summary of results. – KConrad Jun 17 '18 at 4:24

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.