11
$\begingroup$

Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for varying $n$ and in the limit, for $\operatorname{PGL}_2(\mathbb Z_\ell)$?

I would like to know things like the number of irreducible representations, their dimensions, the minimal fields they are defined over and stuff like that.

Improve this question
$\endgroup$
7
  • 2
    $\begingroup$ You might try to look at arxiv.org/pdf/0807.4684.pdf Also, it should be worth to take a look at other related articles of Stasinski. Also, you may find some relevant information in the book of Bushnell--Henniart, "The Local Langlands conjecture for $GL_2$" $\endgroup$
    – AlexIvanov
    Commented Aug 12, 2020 at 19:44
  • 3
    $\begingroup$ The case of $n=p$ is of course well known, e.g., Piatetskii-Shapiro's book. For n squarefree, you can write GL(2,n) as a product of GL(2,p)'s. Similarly, you can reduce to the case of GL(2) over $\mathbb Z/p^r \mathbb Z$, if this helps. $\endgroup$
    – Kimball
    Commented Aug 12, 2020 at 21:25
  • 1
    $\begingroup$ And related for $\mathbb Z_\ell$: mathoverflow.net/q/89184/6518 (I think there are other related questions on this site but can't find them now) $\endgroup$
    – Kimball
    Commented Aug 12, 2020 at 21:26
  • $\begingroup$ Thank you, I will take a look at the references and see if it helps in my case! $\endgroup$
    – Asvin
    Commented Aug 12, 2020 at 21:33
  • 3
    $\begingroup$ You can find further references in the answers to this question: mathoverflow.net/q/87254/2381. For the dimensions and multiplicities for $\mathrm{GL}_2$, Onn's paper is a convenient reference. As far as I know, the case $\mathrm{PGL}_2$ has not been written down (and may require $p>2$ to be manageable). I don't think anyone has studied the minimal fields of definition. $\endgroup$
    – A Stasinski
    Commented Aug 13, 2020 at 10:57

0

Reset to default

You must log in to answer this question.