rt.representation theory - Uniqueness of local Langlands correspondence for connected reductive groups over real/complex field.

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.]