7
$\begingroup$

Let $F$ be a local field of characteristic $0$. Let $D$ be division algebra over $F$ of dimension $n^2$. The construction of irreducible complex representations of $D^*$ is known by Howe, Zink, and others.

I would like to ask the following questions:

What about the construction of mod $p$ representations of $D^*$ for $n\ge 2$?

What is the status of the mod $p$ Jacquet-Langlands correspondence? Are these things explicitly known?

Improve this question
$\endgroup$
6
  • $\begingroup$ Is the residue characteristic of $F$ equals to $p$, or different from $p$? That's two cases which are very different... $\endgroup$
    – Joël
    Commented Jul 31, 2014 at 22:44
  • $\begingroup$ Thanks Joël for pointing two different cases. I am very happy if you suggest some references in both cases. $\endgroup$
    – sampath
    Commented Aug 1, 2014 at 8:14
  • $\begingroup$ Dear user56638, I don't know a reference for any of these cases. The study of mod $p$ representations of reductive groups over a local field $F$ in the case where $p$ is not the residue characteristic of $F$ has been the object of a lot of attention in the 80's, 90's and early 2000's, but I don't know where the case of the group of units of a central simple algebra is treated. You should look carefully in the work of M.-F. Vignéras. $\endgroup$
    – Joël
    Commented Aug 1, 2014 at 18:18
  • $\begingroup$ When $p$ is equal to the residue characteristic of $F$, the mod p representations of $\mathrm{GL}_n(D)$ are studied by Tony Ly in his thesis (Ly). In particular, you can look at Proposition 1.3.1 for the classification in the case $n=1$. $\endgroup$
    – Arkandias
    Commented Aug 9, 2014 at 10:31
  • 1
    $\begingroup$ The mod $p$ representations of $D^*$ are easy to construct. Indeed let $p_D$ be the prime ideal of $D$ and consider the subgroup $U_D^1 =1+p_D$ of $1$-units. It is a pro-$p$-group. If $V$ is an irreducible mod $p$ representation of $D^*$ it has a fixed vector under $U_D^1$. Since $U_D^1$ is normal, $V$ may be viewed as a representation of $D^* /U_D^1$ which is isomorphic to a semi-direct product of $\mathbb Z$ by $k_D^*$, where $k_D$ is the residue field of $D$. It is an easy exercice to work out the irreducible representations of that semidirect product. $\endgroup$
    – Paul Broussous
    Commented Aug 20, 2014 at 12:22

0

Reset to default

You must log in to answer this question.