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.
Best Answer
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.