What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Question

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader knows what the "1970s version of the local Langlands conjectures" are when writing this question---there are plenty of references that will get us this far (I give one below that works in the generality I'm interested in).

So let $F$ be a finite extension of $\mathbf{Q}_p$, let $G$ be a connected reductive group over $F$, let $\widehat{G}$ denote the complex dual group of $G$ (a connected complex Lie group) and let ${}^LG$ denote the $L$-group of $G$, the semi-direct product of the dual group and the Weil group of $F$ (formed using a fixed algebraic closure $\overline{F}$ of $F$).

Here is the "standard", or possibly "standard in the 1970s", way of formulating what local Langlands should say (for more details see Borel's paper "Automorphic $L$-functions", available online (thanks AMS) here at the AMS website. One defines sets $\Phi(G)$ ($\widehat{G}$-conjugacy classes of admissible Weil-Deligne representations from the Weil-Deligne group to the $L$-group [noting that "admissible" includes assertions about images only landing in so-called "relevant parabolics" in the general case and is quite a subtle notion]) and $\Pi(G)$ (isomorphism classes of smooth irreducible admissible representations of $G(F)$), and one conjectures:

LOCAL LANGLANDS CONJECTURE (naive form): There is a canonical surjection $\Pi(G)\to\Phi(G)$ with finite fibres, satisfying (insert list of properties here).

See section 10 of Borel's article for the properties required of the map.

Now in recent weeks I have had two conversations with geometric Langlands type people both of whom have mocked me when I have suggested that this is what the local Langlands conjecture should look like. They point out that studying some set of representations up to isomorphism is a very "coarse" idea nowadays, and one should reformulate things category-theoretically, considering Tannakian categories of representations, and relating them to...aah, well there's the catch. Looking back at what both of them said, they both at a crucial point slipped in the line "well, now for simplicity let's assume we're in the function field/geometric setting. Now..." and off they went with their perverse sheaves. The happy upshot of all of this is that now one has a much better formulation of local Langlands, because one can demand much more than a canonical surjection with finite fibres, one can ask whether two categories are equivalent.

But I have been hoodwinked here, because I am interested in $p$-adic fields. So yes yes yes I'm sure it's all wonderful in the function field/geometric setting, and things have been generalised beyond all recognition. My question is simply:

Q) Can we do better than the naive form of Local Langlands (i.e. is there a stronger statement about two categories being equivalent) when $F$ is a p-adic field?

The answer appears to be "yes" in other cases but I am unclear about whether the answer is yes in the $p$-adic case. Even if someone were to be able to explain some generalisation in the case where $G$ is split, I am sure I would learn a lot. To be honest, I think I'd learn a lot if someone could explain how to turn the surjection into a more bijective kind of object even in the case of $SL(2)$. Even in the unramified case! That's how far behind I am! As far as I can see, the Satake isomorphism gives only a surjection in general, because there is more than one equivalence class of hyperspecial maximal compact in general.

Answer 1

Now that our paper Geometrization of the local Langlands correspondence with Fargues is finally out (ooufff!!), it may be worth giving an update to Ben-Zvi's answer above. In brief: we give a formulation of Local Langlands over a $p$-adic field $F$ so that it is finally

1. an actual conjecture, in the sense that it asks for properties of a given construction, not for a construction;
2. of a form as in geometric Langlands, in particular about an equivalence of categories, not merely a bijection of irreducibles.

First, I should say that in the notation of the OP, we construct a canonical map $\Pi(G)\to \Phi(G)$, and prove some properties about it. However, we are not able to say anything yet about its fibres (not even finiteness).

Moreover, we give a formulation of local Langlands as an equivalence of categories, and (essentially) construct a functor in one direction that one expects to realize the equivalence. In particular, this nails down what the local Langlands correspondence should be, it "merely" remains to establish all the desired properties of it.

Let me briefly state the main result here. Let $\mathrm{Bun}_G$ be the stack of $G$-bundles on the Fargues--Fontaine curve. We define an ($\infty$-)category $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$ of $\ell$-adic sheaves on $\mathrm{Bun}_G$. The stack $\mathrm{Bun}_G$ is stratified into countably many strata enumerated by $b\in B(G)$, and on each stratum, the category $\mathcal D(\mathrm{Bun}_G^b,\overline{\mathbb Q}_\ell)$ is the derived ($\infty$-)category of smooth representations of the group $G_b(F)$. In particular, for $b=1$, one gets smooth representations of $G(F)$.

Moreover, there is an Artin stack $Z^1(W_F,\hat{G})/\hat{G}$ of $L$-parameters over $\overline{\mathbb Q}_\ell$.

Our main result is the construction of the "spectral action":

There is a canonical action of the $\infty$-category of perfect complexes on $Z^1(W_F,\hat{G})/\hat{G}$ on $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$.

The main conjecture is basically that this makes $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^\omega$ a "free module of rank $1$ over $\mathrm{Perf}(Z^1(W_F,\hat{G})/\hat{G})$", at least if $G$ is quasisplit (or more generally, has connected center).

More precisely, assume that $G$ is quasisplit and fix a Borel $B\subset G$ and a generic character $\psi$ of $U(F)$, where $U\subset B$ is the unipotent radical, giving the Whittaker representation $c\text-\mathrm{Ind}_{U(F)}^{G(F)}\psi$, thus a sheaf on $[\ast/G(F)]$, which is the open substack of $\mathrm{Bun}_G$ of geometrically fibrewise trivial $G$-bundles; extending by $0$ thus gives a sheaf $\mathcal W_\psi\in \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$, called the Whittaker sheaf.

Conjecture. The functor $$ \mathrm{Perf}(Z^1(W_F,\hat{G})/\hat{G})\to \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$$ given by acting on $\mathcal W_\psi$ is fully faithful, and extends to an equivalence $$\mathcal D^{b,\mathrm{qc}}_{\mathrm{coh}}(Z^1(W_F,\hat{G})/\hat{G})\cong \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^{\omega}.$$

Here the superscript $\mathrm{qc}$ means quasicompact support, and $\omega$ means compact objects. As $Z^1(W_F,\hat{G})$ is not smooth (merely a local complete intersection), there is a difference between perfect complexes and $\mathcal D^b_{\mathrm{coh}}$, and there is still a minor ambiguity about how to extend from perfect complexes to all complexes of coherent sheaves. Generically over the stack of $L$-parameters, there is however no difference.

It takes a little bit of unraveling to see how this implies more classical forms of the correspondence, like the expected internal parametrization of $L$-packets; in the case of elliptic $L$-parameters, everything is very clean, see Section X.2 of our paper.

(There are related conjectures and results by Ben-Zvi--Chen--Helm--Nadler, Hellmann and Zhu; see also the work of Genestier--Lafforgue in the function field case. And this work is heavily inspired by previous work in geometric Langlands, notably the conjectures of Arinkin--Gaitsgory, and the work of Nadler--Yun and Gaitsgory--Kazhdan--Rozenblyum--Varshavsky on spectral actions.)

PS: It may be worth pointing out that this conjecture is, at least a priori, of a quite different nature than Vogan's conjecture, mentioned in the other answers, which is based on perverse sheaves on the stack of $L$-parameters; here, we use coherent sheaves.

Answer 2

I come back to the original question. First, the title: "What are the local Langlands conjectures nowadays, for connected reductive groups over a p-adic field?" I think they are not far from being proved for classical groups. First, as you know well, they are proved for $GL_n$ over a $p$-adic field. Then, (quasisplit) classical groups can be seen as twisted endoscopic groups of some linear groups, that is to say their $L$-group has a natural representation into some $GL_n (\mathbb{C})$, for some $n$, that allows you to identify your group with a twisted endoscopic group of $GL_n$. Now, thanks to forthcoming work of Arthur (coupled with the recent proof of the twisted weighted fundamental lemma) one will be able to prove Langlands transfert conjecture (a global statement over number fields) for twisted (and untwisted) endoscopy. Using this, coupled with some globalization arguments one will be able to define local transfert (associated to your natural embedding $\;^L G \hookrightarrow GL_n (\mathbb{C})$) from your classical group $G$ toward $GL_n$. You can then define L-paquets as the fibers of this transfert map...well all what I'm saying is vague but it seems to me a big part of this program is carried out in this article

http://people.math.jussieu.fr/~moeglin/paquetgeneral.pdf

Now for the question "Q) Can we do better than the naive form of Local Langlands (i.e. is there a stronger statement about two categories being equivalent) when F is a p-adic field?" the answer is no up to now. First, le me put a comment about the real/p-adic case. There is a big difference between the real Lie groups case and the p-adic fields case: there are no supercuspidals for real groups. The "smallest blocks" of the classification you can identify are discrete series. In the p-adic case you have some "smaller elementary particles" that are supercuspidal representations. The classification of supercuspidals is really arithmetic in nature and I don't see any hope for a geometric Langlands type classification of supercuspidals. All what has been done in geometric Langlands up to my knowledge is to look at objects like affine Grassmanians like $GL_n (k((\pi))/GL_n (k[[\pi]])$ or $GL_n(k((\pi))/I$ with $I$ an Iwahori subgroup of this, nothing with more "depth". There is one geometric thing you can do as explained before: you can pull back Lusztig theory from the finite field case to the depth $0$-part of the representation theory of a p-adic group, but in higher depth there's nothing.

Maybe I should thay this too: assuming you have classified the supercuspidals, an arithmetic task, you can do something geometric that is the following for $GL_n$ over a $p$-adic field $F$. You have your Bernstein center decomposition of the category of representations of $GL_n (F)$, each one is attached to a type in Bushnell-Kutzko sense and they have computed the Hecke algebra of those types: all of them are Iwahori-Hecke algebras. Thus if you have classified all supercuspidals, the representation theory of $GL_n (F)$ goes back to representation theory of Iwahori-Hecke algebras and here you have all the geometric machinery available to work with those category of representations (and you can prove theorems about induced representations of $GL_n(F)$ using this approach, in particular using the Kazhdan-Lusztig conjecture).

A final remark maybe: let $\pi$ be a supercuspidal representation of $GL_n(F)$ and $\sigma(\pi)$ its associated irreducible representation of $W_F$. Suppose $\pi$ is autodual, then $\pi$ is always of orthogonal type (this follows from the fact $\pi$ has a Kirillov model and thus a $1$-dimensional invariant subspace under a compact open subgroup). But $\sigma (\pi)$ may not be orthogonal, it may be symplectic: there's a conjecture by Prasad Ramakrishnan I proved that tells you $\sigma(\pi)$ is orthogonal iff $n$ is odd. Thus, $\pi\mapsto \sigma (\pi)$ is certainly not functorial... The conjecture is in fact more general and involves square integrable representations, in some sens there is something "functorial" that is the following: take $D$ a division algebra with invariant $1/n$ over $F$, then for $\rho$ an irreducible representation of $D^\times$, $$ \rho\otimes JL(\rho)\otimes \sigma_\ell (JL(\rho)) $$ is "canonical" ($\sigma_\ell$ is the $\ell$-adic local Langlands). If there is something categorical to look for, it is hidden behind this...but I'm still looking for it.

Answer 3

A brief update: in his amazing talk yesterday at MSRI (available here), Laurent Fargues explained (building on work of Peter Scholze, see his phenomenal talk two weeks ago here and his historic Berkeley lectures here) a conjectural picture for the local Langlands correspondence as a geometric Langlands correspondence over the "Fargues-Fontaine curve".

[Caveat Emptor: this is way way out of my depth and hopefully will just spur someone knowledgable to comment.] The "curve" is an adic space, and in fact represents a "diamond" as developed in Scholze's Berkeley lectures. This means a functor on perfectoid rings in characteristic p with some nice representability properties. The curve is closely related to the diamond attached to $\mathbb{Q}_p$ itself - the functor which attaches to a perfectoid ring the set of its untilts to characteristic zero.

In any case, Fargues defines the stack $\mathrm{Bun_G}$ of the curve, which looks very very coarsely like $\mathrm{Bun_G}$ of $\mathbb{P}^1$ --- ie it has a (Harder-Narasimhan) stratification where the pieces are classifying spaces of groups, and correspond to isomorphism classes of Kottwitz' G-isocrystals (Frobenius-twisted conjugacy classes in the p-adic group). On the other hand local systems for the dual group on the curve are precisely Langlands parameters. The conjecture is a functor from these local systems to perverse sheaves on $\mathrm{Bun_G}$ which are Hecke eigensheaves --- this notion makes sense thanks to Scholze's new fangled version of the [Beilinson-Drinfeld] affine Grassmannian over $\mathbb{Q}_p$. The functor is required to satisfy other natural analogs of compatibilities from the geometric Langlands program. Fargues ends the talk with various implications of the conjecture, as well as the assertion that it works in the abelian case - ie one should have a purely geometric construction of local class field theory.

Edit: It's important to note that perverse sheaves on $\mathrm{Bun_G}$ of the Fargues-Fontaine curve are closely related to representations of p-adic groups. Namely the semistable locus of $\mathrm{Bun_G}$ (if I understood correctly) is (very roughly) a union of classifying spaces of groups, which are pure inner forms of our group $\mathrm{G}$ over our p-adic field. Hence restricting perverse sheaves to these substacks we obtain a functor to representations. Conversely we might dream of extending sheaves on theses substacks to Hecke eigensheaves on all of $\mathrm{Bun_G}$, and hence attaching to them Langlands parameters. (This might be completely misguided -- but analogous structures do indeed appear in the case of real groups.)

Answer 4

First, I'd like to second the reference given by JT: David Vogan, "The local Langlands conjecture", appearing in Representation Theory of Groups and Algebras (J. Adams et al., eds. Contemporary Mathematics 145. American Mathematical Society, 1993. It can be found on Vogan's webpage

Vogan's article contains a very nice exposition of the local Langlands conjectures, and Arthur's local conjectures, and Vogan's own reformulations which I enjoy. In Conjecture 1.9, Vogan gives the local Langlands conjecture, as the OP has given it. Then, in Conjecture 1.12, Vogan gives a refinement describing L-packets, in the language of perverse sheaves (which the OP may or may not like). Adams, Barbasch, and Vogan proved this refinement for real reductive groups, and Vogan's article is certainly influenced by this.

Later, in Conjecture 4.3, Vogan gives a more detailed version of Langlands original conjectures. In Conjecture 4.15, Vogan gives a refinement, which seems equivalent to some conjectures of Arthur, though I'm not sure. This applies to most cases of interest.

To be specific, regarding $SL_2$ over a $p$-adic field, one must -- in addition to a Weil-Deligne representation $\phi$ into $PGL_2(C)$ -- give an irreducible representation of the component group of the centralizer of the image of $\phi$.

For example, consider an irreducible constituent of an unramified principal series of $SL_2(Q_p)$, whose Weil-Deligne representation $\phi$ sends (geometric, but who cares) Frobenius to the class of a diagonal matrix $diag(-1, 1)$ in $PGL_2(C)$. Note that this matrix is centralized not only by diagonal matrices in $PGL_2(C)$, but also by the Weyl element (since we work in $PGL_2(C)$ and not just $GL_2(C)$). The centralizer of the image of $\phi$ will be the group $N_{\hat G}(\hat T)$ normalizing a maximal torus in $\hat G = PGL_2(C)$, I think. Its component group has order $2$. Since a group of order $2$ has two irreps, there are in fact two irreps of $SL_2(Q_p)$ with this Langlands parameter. This fills out the whole L-packet -- the two irreps occur as constituents in the same principal series in this case.

I think the most helpful treatment of L-packets for $SL_2$ can be found in the recent paper of Lansky and Raghuram, "Conductors and newforms for $SL(2)$", published in Pac. J. of Math, 2007. It's very explicit and considers every case thoroughly, and in a way directly relevant to modular forms. There you can find proven the statements you mention about the nontrivial L-packets being related to the two hyperspecial compact subgroups -- it's also related to the fact that "generic" has two possible meanings for $SL_2$, and representations can be generic for one orbit of character, and not for the other.

Answer 5

To elaborate on Marty's comment, the simple moral one learns from both the Kazhdan-Lusztig classification of tamely ramified representations and the real local Langlands classification is that L-packets are to be expected to be given by representations of component groups of stabilizers (centralizers) of Galois representations, as Vogan conjectures in general. One doesn't need perverse sheaves here -- these representations are the same as equivariant local systems on the appropriate representation variety. In the real case e.g. the variety of Langlands parameters breaks up into a discrete collection of orbits, so we don't see immediately any role for perverse sheaves -- and in any case as far as classification of simple objects (hence irreducible representations) is concerned there's no difference between local systems on disjoint unions of strata and perverse sheaves on an interesting space made out of these strata.

Where things get very rich geometrically is when you try to go beyond classification of irreducibles -- in the real local Langlands story we have that luxury since the first step is already done. How do you go beyond? you might ask for character formulas, relate standard and simple modules, or more ambitiously try to describe the full (derived) category of representations. Adams Barbasch Vogan introduce an interesting space with the same orbit structure as the real Langlands parameters but a much more interesting geometry, and they describe the K-group of representations in terms of equivariant perverse sheaves on this variety, finding a proper geometric context for Vogan's character duality.

In fact one can go much further. Soergel conjectures a real local Langlands classification for the entire derived category of representations (Harish-Chandra modules) of a real group, which lifts Adams-Barbasch-Vogan's picture on K-groups. Roughly speaking this is a derived equivalence between equivariant perverse sheaves on group orbits on flag varieties for Langlands dual groups -- one side gets identified with reps via Beilinson-Bernstein, the other is the ABV Langlands parameters. One can specify this conjecture much more -- it is supposed to be equivariant for intertwining operators/ braid group actions on the two sides, and it has a very particular interaction with t-structures (Koszul duality).

(I would be very interested to learn to what extent one might expect p-adic analogues of any of these more refined versions of local Langlands -- yes, I know, first one might want to prove the original conjectures! - but still it's interesting to dream.)

One cool feature here is both sides look very similar (but for dual groups).. ie once we interpret L-packets as local systems, the Galois side of the correspondence starts to look a lot more like the automorphic side (where we're used to representations being realized in terms of local system type objects on appropriate spaces). In the geometric Langlands setting (which we're supposed not to mention in answers) the two sides really look completely symmetric, and this is the closest indication of that I've seen in the "original" setting.

Answer 6

Surely anything like a categorical equivalence which you seem to be looking for would involve (even to have a statement?) an understanding of the finite fibers of the map you call the naive Local Langlands. As far as I can tell even this is a hard problem: for representations generated by an Iwahori fixed vector (the category studied by Borel) this was solved by Lusztig: in his work with Kazhdan on affine Hecke algebras they show that "L-packets" are given by representations of a certain component group which arise geometrically.

More recently DeBacker and Reeder have given an explicit description of L-packets for "depth zero" supercuspidal representations. One of the key properties they show these L-packets have is a sort of "stability", which seems to me to relate to the representation coming from something "geometric" (perhaps I should say "motivic" -- there's some interesting stuff on the motivic nature of characters by Hales and Gordon?)

I'd also like to compare with the usual toy example of finite groups of Lie type: there the representations (over $\mathbb C$) are classified by Lusztig, and one can describe the classification in terms of data on the dual group, which can largely be interpreted over the complex group. The notion of stability again arises when you study base change: trying to match representations of $G(F_q)$ with certain representations of $G(F_{q^n})$. In general this can only be done if you package together representations into things like which could be called L-packets (which are explicitly understood). The process of doing so matches up with the process of understanding how the category of representations of the groups $G(F_q)$ compare with the category of character sheaves on the group (the "geometric" category). Even in this toy case though I don't know a nice categorical way of saying how the two are related.

Answer 7

Here's a remark about the Satake isomorphism, although I don't think it's what you're looking for: Suppose $G$ is a split reductive group over $\mathbb{Z}$, then one formulation of Satake is an isomorphism

$H(G(\mathbb{Q}_p),G(\mathbb{Z}_p)) \rightarrow \mathbb{C}\otimes K(Rep(\hat{G}))$

where $H(G(\mathbb{Q}_p),G(\mathbb{Z}_p))$ is the Hecke algebra with respect to the maximal compact $G(\mathbb{Z}_p)$, and $K(Rep(\hat{G}))$ is the Grothendieck ring of the category of finite dimensional reps of the Langlands dual group $\hat{G}$ (a reductive group over $\mathbb{C}$).

But now write $K$ for $\mathbb{F}_p((T))$, $A$ for $\mathbb{F}_p[[T]]$. We also have a Satake isomorphism for the Hecke algebra $H(G(K),G(A))$, with exactly the same target as our earlier isomorphism! So the Satake isomorphism doesn't seem to see any difference between p-adic fields and function fields, and indeed one can view the categorified Satake isomorphism of, say, Mirkovic-Vilonen as saying something for the p-adic field case as well.

Answer 8

This is not an answer but a followup question. In the p-adic case, is there any hope for a set of conditions on the local Langlands correspondence which would make it unique? In the case of $GL(n)$ this is provided by L and epsilon factors. For classical groups, can one make a precise statement (even conjecturally) for classical groups, using lifting to $GL(n)$?