1
$\begingroup$

I am wondering why the first well known example of non-tempered irreducible admissible representation of $p$-adic group $U(n)$ should be $U(3)$. Because, Gelbart and Rogawski suggested the non-tempered representation of $U(3)$ using the theta lift.

I know all irreducible representation of $U(1)$ should be tempered because $U(1)$ is compact. But why isn't there known example for the non-tempered representation of $U(2)$? I think it should be there.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

The admissible representations of U(2) over a p-adic field are fairly straightforward since they essentially come from those of SL(2) or a the norm one elements of the quaternions.

Let me add (two days later) that the point is not that the representation of U(3) that arises from the theta-lift is non-tempered but rather that is non-tempered AND the way it arises from a theta-lift. This allows you to construct CAP representations of U(3) globally.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thank you for your comment. Then do you mean all irreducible admissible representation of U(2) is tempered representation? $\endgroup$
    – Monty
    Commented Feb 23, 2015 at 5:25
  • $\begingroup$ No. Not all irreducible admissible representations of SL(2) are tempered. $\endgroup$
    – mander
    Commented Feb 23, 2015 at 13:21
  • $\begingroup$ Thanks for your kind explantion. It helped me very much! $\endgroup$
    – Monty
    Commented Mar 1, 2015 at 3:49

You must log in to answer this question.

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