2
$\begingroup$

Recently, I came across the Langlands correspondence theorem, there is the following line:

$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$ where $\sigma$ and $\tau$ are representations of the Weil group, and $\pi(\sigma)$ and $\pi(\tau)$ are representations of $GL_2(F)$.

My question is: Why is the first symbol '$\times$' while the second symbol is '$\otimes$'? Are these interchangeable?

Actually I have seen $L(s,\pi \otimes \chi)$ and $L(s,\pi \otimes f)$ (with $\pi$ representation, $\chi$ character, and $f$ modular form) in some places, but in other literature, I have seen authors write them as $L(s,\pi \times \chi)$ and $L(s,\pi \times f)$. I'm really confused.

Improve this question
$\endgroup$
0

1 Answer 1

Reset to default
5
$\begingroup$

The symbol $\times$ on the left-hand side is the Rankin-Selberg product. If $\pi$ and $\rho$ are automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(n)$, respectively, then one can define the Rankin-Selberg $L$-function $L(s,\pi\times\rho)$. Conjecturally, there is an automorphic representation of $\mathrm{GL}(mn)$ whose $L$-function is $L(s,\pi\times\rho)$, and $\pi\times\rho$ is supposed to denote this automorphic representation. For $m=n=2$ the conjecture was proved by Ramakrishnan (2000), so $\pi(\sigma)\times\pi(\tau)$ in the original post is a genuine automorphic representation of $\mathrm{GL}(4)$. If $\pi\times\rho$ is known to exist, then it is reasonable to denote it by $\pi\otimes\rho$. In particular, this is the case when $\rho$ is a character ($n=1$): in this case, $\pi\otimes\rho$ is the automorphic representation $g\mapsto\pi(g)\rho(\det g)$.

The symbol $\otimes$ on the right-hand side is the usual tensor product of group representations.

Improve this answer
$\endgroup$
6
  • $\begingroup$ Thank you very much! So all '$\otimes$' can be replaced with '$\times$', right? (In fact I have seen people do this.) $\endgroup$
    – Misaka 16559
    Commented Dec 21, 2023 at 14:21
  • 1
    $\begingroup$ @EulerLee For Galois (or Weil-Deligne) representations, I would stick to $\otimes$. For automorphic representations, $\times$ is a safe and good option. $\endgroup$
    – GH from MO
    Commented Dec 21, 2023 at 14:49
  • 2
    $\begingroup$ I use $\otimes$ on the automorphic side too (at least when it's known to be automorphic). You might see, e.g., $\pi \times \rho$ for a rep of say GL(2) x GL(2) and $\pi \otimes \rho$ for the functorial transfer to GL(4). $\endgroup$
    – Kimball
    Commented Dec 21, 2023 at 15:34
  • $\begingroup$ Isn't there a risk of confusion with the actual tensor product of the representations of GL(2) (maybe completed)? I guess this one might not be irreducible nor automorphic. $\endgroup$
    – Aurel
    Commented Dec 21, 2023 at 16:41
  • 2
    $\begingroup$ @GHfromMO Fair enough! $\endgroup$
    – Aurel
    Commented Dec 21, 2023 at 17:09

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .