How badly can strong multiplicity one fail in the theory of automorphic representations?

Question

Let $G$ be a connected reductive group over a global field $k$, and let $\pi=\otimes_w\pi_w$ and $\pi'=\otimes_w\pi'_w$ be two automorphic representations for $G$, where here of course $w$ is ranging over all the places of $k$.

Assume now that $\pi_w\cong\pi'_w$ for all but finitely many places $w$ of $k$. I think people say "$\pi$ and $\pi'$ are nearly equivalent".

If ($G=GL(n)$ and $\pi$ is cuspidal), or if ($G=GL(n)$ and $\pi$ and $\pi'$ occur discretely in $L^2$), then this would force $\pi_v\cong\pi'_v$ for all places $v$. But in general this "strong multiplicity one" phenomenon does not occur. Indeed even if $G=GL(2)$ we can have $\pi_v\not\cong\pi'_v$ for a non-zero finite set of $v$: if $\pi$ is 1-dimensional then $\pi'$ can be Steinberg at $v$, for example. For groups other than $GL(2)$ we can even have $\pi$ and $\pi'$ cuspidal, with $\pi_v\not\cong\pi'_v$ -- this even happens if $G=SL(2)$: "strong multiplicity one" can fail here.

So here's a vague question. We've established that $\pi$ and $\pi'$ nearly equivalent does not imply $\pi_v\cong\pi'_v$ for all $v$. But can we say anything about the relationship between $\pi_v$ and $\pi'_v$?

But I am not a fan of vague questions so here are some more precise ones, together with some guesses for answers. Say $\pi_w\cong\pi'_w$ for almost all $w$, but $\pi_v\not\cong\pi'_v$.

0) Do $\pi_v$ and $\pi'_v$ necesarily have the same central character? [this should be an easy warm-up. It's just the question of whether tori satisfy some sort of strong mult 1. I feel a bit lame not being able to figure this out :-/]

1) Are $\pi_v$ and $\pi'_v$ necessarily in the same Bernstein component? [my guess is "no"; I half-suspect that for $G=GSp(4)$ one can have $\pi_v$ supercuspidal and $\pi'_v$ not, but my source is "I think someone once told me this" and it would be nice to have a more concrete one].

2) If $v$ is infinite, do $\pi_v$ and $\pi'_v$ have the same infinitesimal character? [My guess is "this is known for $GL(n)$, and might follow from a super-optimistic version of Langlands functoriality for general $G$ but perhaps I am being a bit too optimistic."]

3) If $v$ is infinite, are $\pi_v$ and $\pi'_v$ in the same local $L$-packet as defined by Langlands? [I have very little understanding of local $L$-packets at infinity and daren't hazard a guess.]

4) Back to general $v$. Should one expect that $\pi_v$ and $\pi'_v$ are in the same "packet" in some way? I write this in quotes because I don't know that I can give a definition of $L$-packet or $A$-packet in this generality. So here I daren't even have an opinion.

I'd be interested to know in anything that is proved or conjectured.

Answer 1

I confirm what "someone once told you" about question 1 (so now "two people once told you" or perhaps "someone twice told you"). This phenomenon ($\pi_\nu$ supercuspidal, and $\pi'_\nu$ principal series, even unramified) occurs for example when $\pi$ and $\pi'$ are the non-tempered endoscopic representation in the discrete (or even cuspidal) spectrum that some people like to deform, for $U(3)$ and $GSP(4)$ and their inner forms. There is an article by Rogawski, "The multiplicity formula for A-packets" in the book "the Zeta Function of Picard Modular Surfaces", where he describes in details such an example for each of the two inner forms of $U(3)$ attached to a quadratic imaginary field $E$. You have analog examples for $GSP_4$, where $\pi$ is a Saito-Kurokawa lift of a modular form.

Answer 2

Regarding relationships between 0), 2), 3), and 4) (but not really an answer to any of them without additional insights) :

a) It's predicted (see Borel's Corvallis article page 44) that if two representations belong to the same local "$L$-packet", then they have the same central character.

b) Let $G(\mathbb{R})$ be a real group that has relative discrete series representations. Fix a central character $\chi$ and an infinitesimal character $\lambda$. Then the set of relative discrete series representations of $G(\mathbb{R})$ with central character $\chi$ and infinitesimal character $\lambda$ is an $L$-packet of $G(\mathbb{R})$, and every relative discrete series $L$-packet of $G(\mathbb{R})$ is of this form.

c) Icing on whatever cake exists : For explicit constructions of central characters associated to Langlands parameters : Given a discrete Langlands parameter for a real group $G(\mathbb{R})$, Langlands has constructed the central character of the associated $L$-packet. If $G(F)$ is $p$-adic such that the maximal torus in its center is anisotropic, Gross/Reeder (Section 8 of their recent paper for this, which also contains Langlands' above construction for real groups) have constructed a central character attached to a discrete Langlands parameter for $G(F)$. fwiw : Borel (in Corvallis) gave a construction of a central character attached to a Langlands parameter of an arbitrary connected reductive group over a real or $p$-adic field $k$, but it is noted in the first paragraph of Section 8 Gross/Reeder that Borel's construction "omits an essential point, namely the vanishing of the Schur multiplier of $Gal(\overline{k} / k)$, due to Tate".

Answer 3

They are in the same Arthur packet. The A-packets are designed exactly to answer this question. I think the answers to questions 1,2,3 are "no", "yes conjecturally", and "no": see book of Adams Barbasch Vogan or Vogan's survey article on the local Langlands conjectures. Note that, although the A-packet is not a union of L-packets in general, it at least contains a canonical L-packet.

[Oops: answer edited. This only applies to "automorphic" in the sense of "appearing discretely in the L^2-spectrum". Misunderstood the focus of question. Explicit reference for 1 is e.g. 1.11 of Gan-Gurevich "Non-tempered A-packets of G_2: liftings from SL_2-tilde."]

Answer 4

For what it's worth I can now answer Q0. I believe it is not true in general that $\pi_v$ and $\pi'_v$ will have to have the same central character. We can let $G$ be a torus $T$. If $S$ is a finite set of places of $k$ then we can set $\pi_w=\pi'_w=1$ for $w\not\in S$. In fact let's set $\pi'=1$. Can we find $\pi$ with $\pi\not=1$? This just boils down to the question of whether $T(k)$ is dense in $\prod_{v\in S}T(k_v)$. This is "bien connu" not to be true in general, according to the first paragraph of this paper of Colliot-Thelene and Suresh available here:

Skorobogatov tells me that Milne's Arithmetic Duality Theorems is the place to look for this well-known stuff. Thanks to Ambrus Pal for pointing out the C-T--S paper.

But even better -- the paper linked to above even answers, in the function field case, the stronger question of whether the induced map $T(k)\to\prod_{v\in S}T(k_v)/M$ is surjective, where $M$ is the maximal compact subgroup -- even this may fail. This means, I believe, that even tori give examples where $\pi_v$ and $\pi'_v$ are not in the same Bernstein component, so we get another counterexample to (1). Note however that it only applies in the function field setting (this is where the C-T--S counterexample takes place).