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

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.

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 Excellent question, I would like to see an answer too. My impression is that the answer is “Not yet”, but probably I am wrong. It is based on the fact that I haven’t seen or heard it anywhere, and I have been looking. – MBN Feb 9 2010 at 14:15 You might like to look at David Vogan's article here: www-math.mit.edu/~dav/md.pdf At least, it gives a formulation of the conjectures a decade or two older than the one you have in mind! – JT Feb 9 2010 at 15:10

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.

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 Laurent - thanks for a very enlightening answer! Certainly I wouldn't expect a geometric classification of supercuspidals.. but one can still ask for a geometric (categorical?) description one Bernstein component at a time (as Marty suggests), like we have in the tamely ramified case.. for GL_n I guess this follows from type theory as you explained? There is now a local geometric Langlands program for arbitrary depth.. it's rather formal right now except in some special cases, but there is a precise general conjecture (the tamely ramified case is the theorem of Bezrukavnikov McGerty mentions) – David Ben-Zvi Feb 10 2010 at 16:00 David - Of course I did not mean you expect a geometric classification of supercuspidals, maybe I misspoke, sorry for this. I just wanted to point the big difference between the p-adic and the real case that makes the real case more manageable with the geometric methods. I did not know their is a precise conjecture for geometric Langlands in any depth, thanks for this. Just a question: is it proved for $GL_1$ (by purely local methods, without globalizing to a smooth projective curve) ? Can you link this to Serre's geometric class field theory ? – Laurent F. Feb 10 2010 at 20:04 I haven't thought carefully about the abelian case (or others really!), but I imagine it should be an application of the Contou-Carrere self-duality of the Jacobian over the disc - a kind of 2-categorical Fourier-Mukai transform for it - which is also behind geometric CFT.. something to think about! – David Ben-Zvi Feb 12 2010 at 4:44

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.

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"...perverse sheaves (which the OP may or may not like)". Aah you misunderstand. What I was objecting to was not the perverse sheaves, but the fact that the perverse sheaves were always on a space that was introduced just after the fateful words "now let's assume we're in an equicharacteristic setting". On the other hand I see that Vogan introduces a space in the p-adic setting too, so I am cautiously optimistic... – Kevin Buzzard Feb 9 2010 at 22:46
I did misunderestimate :) - sorry! Vogan's treatment is the only one - or the first one, at least - to use perverse sheaves in the p-adic setting. As Ben-Zvi states below, it seems like the best way to understand not only the irreducibles, but also the structure of the category. From my perspective, Bernstein's center produces a complex variety, which "roughly" parameterizes representations. For each component in this variety, one should do something like Vogan to study the category of representations supported on this component. – Marty Feb 10 2010 at 7:48

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.

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 Hey Kevin. I was wondering whether somehow the understanding of the packets would somehow be swallowed up by other "finite errors" when one 'categorified'. Explicitly: the L-packets have size 1 when G=GL_n. But it's not hard to find examples of group homs r1,r2:W-->L (think L the L-group, W a Weil group) which have the property that r1(w) and r2(w) are conjugate for all w, but for which r1 isn't conjugate to r2. That's some sort of "finite error" which would be somehow swallowed up if one considered reps L-->GL_n, where that phenomenon doesn't occur. Does Ka-Lu apply even in the p-adic case? – Kevin Buzzard Feb 9 2010 at 16:27 Hey, that's a cool idea about L-packets! I wonder if there's a way to see if it squares with what people already know? The Ka-Lu thing was about p-adic groups: via the Borel idea the category of Iwahori reps is equivalent to the category of finite dimensional Hecke algebra reps and then Ka-Lu just go and classify these. Much more recently, this has been "categorified" by Bezrukavnikov and collaborators, but the categorifications only have to do with geometric/function field cases. I suppose you view that as an extension of James's comment about Satake also (going from G(O) to I). – Kevin McGerty Feb 9 2010 at 17:45 I think there are general conjectures relating sizes of $L$-packets to representation-theoretic data but I am unclear as to what they are. I think that perhaps locally there are precise ones but globally, at least in the SL_2 case, there is some sort of product formula and some subtleties. I am sufficiently ignorant about this to be unable to make precise statements, but I'm pretty sure that at least the local part of the story is well-understood to a certain extent (just not by me). – Kevin Buzzard Feb 10 2010 at 7:35

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.

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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.

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Heh, so far I have two answers, one from the guy in the office down the corridor and one from someone I strongly suspect is one of my graduate students :-) OK so here's my question in this setting. Consider the case G=SL_n. Then there is "more than once choice for G(Z_p)", i.e. there's more than one conj class of hyperspecial max compact. This is one source of packets of size > 1 in this setting. So one basic sanity check one could do on this "lead" would be to see if there's any chance that it could possibly ever lead to an explanation of this phenomenon. I'm printing out the M-V paper... – Kevin Buzzard Feb 9 2010 at 16:23
Yes, it was amusing for a while when the only answers were from three people in the same department. – Emerton Feb 10 2010 at 4:23

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)?

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This might arguably get better viewing figures if it were asked as a separate question. – Kevin Buzzard May 11 2010 at 18:03
I think that the answer is yes, but a comment box is not the place to try to explain more! – Emerton May 11 2010 at 18:04
I made this a separate question. – Jeffrey Adams May 12 2010 at 12:01