4
$\begingroup$

What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$?

One knows such a representations cannot be smooth, so probably the examples will be contrived, but so be it.

$\endgroup$
4
  • 5
    $\begingroup$ Well, there are some obvious $p$-adic reps of $SL_2(\mathbb{Q}_p)$ (namely $Sym^n$ of the standard rep); now choose an embedding of $\mathbb{Q}_p$ into $\mathbb{C}$... $\endgroup$ Jul 17, 2019 at 12:19
  • 4
    $\begingroup$ See an answer to mathoverflow.net/questions/259491/… $\endgroup$
    – Bugs Bunny
    Jul 17, 2019 at 12:26
  • 2
    $\begingroup$ Thanks Daniel. Is it obvious that these representations stay irreducible after applying the embedding? $\endgroup$
    – Spinoza
    Jul 17, 2019 at 12:37
  • 2
    $\begingroup$ @Spinoza yes — the reps are irreducible over $\overline{\mathbb{Q}_p}$. Now the Lemma you need is that irreducibility is preserved under extensions of algebraically closed fields. $\endgroup$ Jul 17, 2019 at 14:41

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.