# Uniqueness of local Langlands correspondence for connected reductive groups over real/complex field.

In Langlands' notes "On the classification of irreducible representations of real algebraic groups", available at the Langlands Digital Archive page here, Langlands gives a construction which is now referred to as "the local Langlands correspondence for real/complex groups".

What Langlands does in practice in this paper is the following. Let $K$ denote either the real numbers or a finite field extension of the real numbers (for example the complex numbers, or a field isomorphic to the complex numbers but with no preferred isomorphism). Let $G$ be a connected reductive group over $K$. Langlands defines two sets $\Pi(G)$ (infinitesimal equivalence classes of irreducible admissible representations of $G(K)$) and $\Phi(G)$ ("admissible" homomorphisms from the Weil group of $K$ into the $L$-group of $G$, modulo inner automorphisms). He then writes down a surjection from $\Pi(G)$ to $\Phi(G)$ with finite fibres, which he constructs in what is arguably a "completely canonical" way (I am not making a precise assertion here). Langlands proves that his surjection, or correspondence as it would now be called, satisfies a whole bunch of natural properties (see for example p44 of "Automorphic $L$-functions" by Borel, available here), although there are other properties that the correspondence has which are not listed there---for example I know a statement explaining the relationship between the Galois representation attached to a $\pi$ and the one attached to its contragredient, which Borel doesn't mention, but which Jeff Adams tells me is true, and there is another statement about how infinitesimal characters work which I've not seen in the literature either.

So this raises the following question: is it possible to write down a list of "natural properties" that one would expect the correspondence to have, and then, crucially, to check that Langlands' correspondence is the unique map with these properties? Uniqueness is the crucial thing---that's my question.

Note that the analogous question for $GL_n$ over a non-arch local field has been solved, the crucial buzz-word being "epsilon factors of pairs". It was hard work proving that at most one local Langlands correspondence had all the properties required of it---these properties are listed on page 2 of Harris-Taylor's book and it's a theorem of Henniart that they suffice. The Harris-Taylor theorem is that at least one map has the required properties, and the conclusion is that exactly one map does. My question is whether there is an analogue of Henniart's theorem for an arbitrary connected reductive group in the real/complex case.

-
No. My instinct a year ago would be to ask an expert. My instinct now is to post here first and then ask an expert if nothing is forthcoming. Tony Scholl pointed out to me by email that Henniart's theorem in the p-adic case isn't for any one specific GL_n, as it were, it's a set of conditions that characterise all of the bijections for all GL_n at once. So perhaps asking for properties that characterise the correspondence for one G is not the right idea and one should do all G at once. – Kevin Buzzard Jul 10 '10 at 19:55
Ah, I had assumed in your question that you were also thinking to work with all $G$ at once, for exactly the kind of reason you mention in your comment. (In the GL_n story there's the extra wrinkle that the GL_1 case has to be put in "by hand", via CFT.) Perhaps the "safest" bet is to work throughout with "split" groups over the reals on a first pass through the question, since in principle that permits a lot more use of inductive technique with many torus centralizers to hope to build things up (somehow) from GL_1 and SL_2. – BCnrd Jul 12 '10 at 12:43

Dear Kevin,

Here are some things that you know.

(1) Every non-tempered representation is a Langlands quotient of an induction of a non-tempered twist of a tempered rep'n on some Levi, and this description is canonical.

(2) Every tempered rep'n is a summand of the induction of a discrete series on some Levi.

(3) The discrete series for all groups were classified by Harish-Chandra.

Now Langlands's correspondence is (as you wrote) completely canonical: discrete series with fixed inf. char. lie in a single packet, and the parameter is determined from the inf. char. in a precise way.

All the summands of an induction of a discrete series rep'n are also declared to lie in a single packet. So all packet structure comes from steps (1) and (2).

The correspondence is compatible in a standard way with twisting, and with parabolic induction.

So:

If we give ourselves the axioms that discrete series correspond to irred. parameters, that the correspondence is compatible with twisting, that the correspondence is compatible with parabolic induction, and that the correspondence is compatible with formation of inf. chars., then putting it all together, it seems that we can determine step 1, then 2, then 3.

I don't know if this is what you would like, but it seems reasonable to me.

Why no need for epsilon-factor style complications: because there are no supercuspidals, so everything reduces to discrete series, which from the point of view of packets are described by their inf. chars. In the p-adic world this is just false: all the supercuspidals are disc. series, they have nothing analogous (at least in any simple way) to an inf. char., and one has to somehow identify them --- hence epsilon factors to the rescue.

[Added: A colleague pointed out to me that the claim above (and also discussed below in the exchange of comments with Victor Protsak) that the inf. char. serves to determine a discrete series L-packet is not true in general. It is true if the group $G$ is semi-simple, or if the fundamental Cartan subgroups (those which are compact mod their centre) are connected. But in general one also needs a compatible choice of central character to determine the $L$-packet. In Langlands's general description of a discrete series parameter, their are two pieces of data: $\mu$ and $\lambda_0$. The former is giving the inf. char., and the latter the central char.]

-
I didn't know all of this Emerton, but it does look good. In fact if you like I'll mention an explicit example of something I'd not really realised until this post. I am pretty sure that it's true that for a general representation, the inf char was too weak to read off the parameter. For example, consider a reducible princ series rep for GL_2(R). This splits up into a f.d. piece and a discrete series piece. These two pieces, if I'm not mistaken, will have the same inf char and rather different associated Weil reps. This caused me some grief at some point when figuring out the details of... – Kevin Buzzard Jul 15 '10 at 10:00
...my almost-finished paper with Toby Gee about auto reps and Galois reps in the global setting. But in fact what you're saying perhaps is that if I restrict to discrete series reps then this issue does not occur and the inf char is a stronger invariant than I had realised---and of course now you phrase it this way I guess it's somehow clear from what Harish-Chandra did. – Kevin Buzzard Jul 15 '10 at 10:02
I don't think that the bit about discrete series being determined by their infinitesimal characters is correct: the Harish-Chandra parameters of a d.s. representation is a $K$-dominant weight, hence there are $W_G/W_K$ discrete series representations of $G$ with the same inf char (this is even mentioned at the en.wikipedia.org/wiki/Discrete_series_representation Wikipedia article). You can see it already in the case of $SL_2(\mathbb{R}):$ holomorphic and antiholomorphic discrete series reps have the same inf char. – Victor Protsak Jul 15 '10 at 17:28
Dear Victor, The claim is not that discrete series are determined by their inf. char; as you note, that is false. But all these discrete series with the given inf. char. lie in the same $L$-packet, essentially by definition. So the Langlands parameter of a disc. series depends only on the inf. char. – Emerton Jul 15 '10 at 17:37
Dear Emerton, I don't think your post implied it, but that's how I read Kevin's response right below it. I may be a bit rusty with my Langlands parameters, because I thought that a Langlands packet may contain at most one discrete series rep, whereas, as you say, a packet consists of $\textit{all}$ d.s. representations with the same inf character. – Victor Protsak Jul 16 '10 at 4:26

Here is a slight refinement of Emerton's answer.

Any L-homomorphism $\phi:W_\mathbb R\rightarrow \phantom{}^LG$ defines an infinitesimal character $\lambda$ and a central character $\chi$. If $\phi(W_\mathbb R)$ is not contained in a proper Levi this defines an L-packet: the set of relative discrete series with this infinitesimal and central character (relative = discrete series modulo center). Every relative discrete series L-packet arises this way.

Now compatibility with parabolic induction determines all L-packets as follows. If $\phi(W_\mathbb R)$ is contained in a proper Levi subgroup $\phantom{}^LM$, the preceding construction defines a relative discrete series L-packet for $M$. The L-packet for $G$ is defined to be the irreducible summands of $Ind_{MN}^G(\pi_M)$ as $\pi_M$ runs over the L-packet of $M$ (for the correct choice of $N$). This induction can be done in stages: a tempered step, which is completely reducible, followed by an induction which gives unique irreducible summands. (This incorporates the "twisting" mentioned above).

As Emerton says this is canonical since we don't need L or epsilon factors to define relative discrete series L-packets. I'd like to know if anything like this, however speculative, might hold for general p-adic groups; in particular whether epsilon factors really do come to the rescue outside of GL(n) and perhaps classical groups. See Characterizing the Local Langlands Correspondence.

-